Limit inferior and limit superior

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https://en.wikipedia.org/wiki/Limit_inferior_and_limit_superior

In mathematics, the limit inferior and limit superior of a sequence can be thought of as limiting (that is, eventual and extreme) bounds on the sequence. They can be thought of in a similar fashion for a function (see limit of a function). For a set, they are the infimum and supremum of the set's limit points, respectively. In general, when there are multiple objects around which a sequence, function, or set accumulates, the inferior and superior limits extract the smallest and largest of them; the type of object and the measure of size is context-dependent, but the notion of extreme limits is invariant. Limit inferior is also called infimum limit, limit infimum, liminf, inferior limit, lower limit, or inner limit; limit superior is also known as supremum limit, limit supremum, limsup, superior limit, upper limit, or outer limit.

An illustration of limit superior and limit inferior. The sequence x_n is shown in blue. The two red curves approach the limit superior and limit inferior of x_n, shown as dashed black lines. In this case, the sequence accumulates around the two limits. The superior limit is the larger of the two, and the inferior limit is the smaller. The inferior and superior limits agree if and only if the sequence is convergent (i.e., when there is a single limit).

The limit inferior of a sequence ${\displaystyle (x_n)}$ is denoted by

$${\displaystyle \underset{n\to \mathrm{\infty }}{lim inf}{x}_n\text{or}\underset{n\to \mathrm{\infty }}{\underset{\_}{\lim }}{x}_n,}$$

and the limit superior of a sequence ${\displaystyle (x_n)}$ is denoted by

$${\displaystyle \underset{n\to \mathrm{\infty }}{lim sup}{x}_n\text{or}\underset{n\to \mathrm{\infty }}{\overline{\lim }}{x}_n.}$$

Definition for sequences

The limit inferior of a sequence (x_n) is defined by

$${\displaystyle \underset{n\to \mathrm{\infty }}{lim inf}{x}_n:=\underset{n\to \mathrm{\infty }}{lim}(\underset{m\ge n}{inf}{x}_m)}$$

or

$${\displaystyle \underset{n\to \mathrm{\infty }}{lim inf}{x}_n:=\underset{n\ge 0}{sup}\underset{m\ge n}{inf}{x}_m=sup\{inf\{x_m:m\ge n\}:n\ge 0\}.}$$

Similarly, the limit superior of (x_n) is defined by

$${\displaystyle \underset{n\to \mathrm{\infty }}{lim sup}{x}_n:=\underset{n\to \mathrm{\infty }}{lim}(\underset{m\ge n}{sup}{x}_m)}$$

or

$${\displaystyle \underset{n\to \mathrm{\infty }}{lim sup}{x}_n:=\underset{n\ge 0}{inf}\underset{m\ge n}{sup}{x}_m=inf\{sup\{x_m:m\ge n\}:n\ge 0\}.}$$

Alternatively, the notations ${\displaystyle \underset{n\to \mathrm{\infty }}{\underset{\_}{\lim }}{x}_n:=\underset{n\to \mathrm{\infty }}{lim inf}{x}_n}$ and ${\displaystyle \underset{n\to \mathrm{\infty }}{\overline{\lim }}{x}_n:=\underset{n\to \mathrm{\infty }}{lim sup}{x}_n}$ are sometimes used.

The limits superior and inferior can equivalently be defined using the concept of subsequential limits of the sequence ${\displaystyle (x_n)}$. An element ${\displaystyle \xi }$ of the extended real numbers ${\displaystyle \overline{\mathbb{R}}}$ is a subsequential limit of ${\displaystyle (x_n)}$ if there exists a strictly increasing sequence of natural numbers ${\displaystyle (n_k)}$ such that ${\displaystyle \xi =\underset{k\to \mathrm{\infty }}{lim}{x}_{n_k}}$. If ${\displaystyle E\subseteq \overline{\mathbb{R}}}$ is the set of all subsequential limits of ${\displaystyle (x_n)}$, then

${\displaystyle \underset{n\to \mathrm{\infty }}{lim sup}{x}_n=supE}$

and

${\displaystyle \underset{n\to \mathrm{\infty }}{lim inf}{x}_n=infE.}$

If the terms in the sequence are real numbers, the limit superior and limit inferior always exist, as the real numbers together with ±∞ (i.e. the extended real number line) are complete. More generally, these definitions make sense in any partially ordered set, provided the suprema and infima exist, such as in a complete lattice.

Whenever the ordinary limit exists, the limit inferior and limit superior are both equal to it; therefore, each can be considered a generalization of the ordinary limit which is primarily interesting in cases where the limit does not exist. Whenever lim inf x_n and lim sup x_n both exist, we have

${\displaystyle \underset{n\to \mathrm{\infty }}{lim inf}{x}_n\le \underset{n\to \mathrm{\infty }}{lim sup}{x}_n.}$

The limits inferior and superior are related to big-O notation in that they bound a sequence only "in the limit"; the sequence may exceed the bound. However, with big-O notation the sequence can only exceed the bound in a finite prefix of the sequence, whereas the limit superior of a sequence like e⁻ⁿ may actually be less than all elements of the sequence. The only promise made is that some tail of the sequence can be bounded above by the limit superior plus an arbitrarily small positive constant, and bounded below by the limit inferior minus an arbitrarily small positive constant.

The limit superior and limit inferior of a sequence are a special case of those of a function (see below).

The case of sequences of real numbers

In mathematical analysis, limit superior and limit inferior are important tools for studying sequences of real numbers. Since the supremum and infimum of an unbounded set of real numbers may not exist (the reals are not a complete lattice), it is convenient to consider sequences in the affinely extended real number system: we add the positive and negative infinities to the real line to give the complete totally ordered set [−∞,∞], which is a complete lattice.

Interpretation

Consider a sequence ${\displaystyle (x_n)}$ consisting of real numbers. Assume that the limit superior and limit inferior are real numbers (so, not infinite).

• The limit superior of ${\displaystyle x_n}$ is the smallest real number ${\displaystyle b}$ such that, for any positive real number ${\displaystyle \epsilon }$, there exists a natural number ${\displaystyle N}$ such that ${\displaystyle x_n<b+\epsilon }$ for all ${\displaystyle n>N}$. In other words, any number larger than the limit superior is an eventual upper bound for the sequence. Only a finite number of elements of the sequence are greater than ${\displaystyle b+\epsilon }$.
• The limit inferior of ${\displaystyle x_n}$ is the largest real number ${\displaystyle b}$ such that, for any positive real number ${\displaystyle \epsilon }$, there exists a natural number ${\displaystyle N}$ such that ${\displaystyle x_n>b-\epsilon }$ for all ${\displaystyle n>N}$. In other words, any number below the limit inferior is an eventual lower bound for the sequence. Only a finite number of elements of the sequence are less than ${\displaystyle b-\epsilon }$.

Properties

In case the sequence is bounded, for all ${\displaystyle ϵ>0}$ almost all sequence members lie in the open interval ${\displaystyle (\underset{n\to \mathrm{\infty }}{lim inf}{x}_n-ϵ,\underset{n\to \mathrm{\infty }}{lim sup}{x}_n+ϵ).}$

The relationship of limit inferior and limit superior for sequences of real numbers is as follows:

$${\displaystyle \underset{n\to \mathrm{\infty }}{lim sup}\left(-x_n\right)=-\underset{n\to \mathrm{\infty }}{lim inf}{x}_n}$$

As mentioned earlier, it is convenient to extend ${\displaystyle \mathbb{R}}$ to ${\displaystyle [-\mathrm{\infty },\mathrm{\infty }].}$ Then, ${\displaystyle \left(x_n\right)}$ in ${\displaystyle [-\mathrm{\infty },\mathrm{\infty }]}$ converges if and only if

$${\displaystyle \underset{n\to \mathrm{\infty }}{lim inf}{x}_n=\underset{n\to \mathrm{\infty }}{lim sup}{x}_n}$$

in which case ${\displaystyle \underset{n\to \mathrm{\infty }}{lim}{x}_n}$ is equal to their common value. (Note that when working just in ${\displaystyle \mathbb{R},}$ convergence to ${\displaystyle -\mathrm{\infty }}$ or ${\displaystyle \mathrm{\infty }}$ would not be considered as convergence.) Since the limit inferior is at most the limit superior, the following conditions hold

$${\displaystyle \begin{array}{rlrl}\underset{n\to \mathrm{\infty }}{lim inf}{x}_n& =\mathrm{\infty }& & \text{ implies }\underset{n\to \mathrm{\infty }}{lim}{x}_n=\mathrm{\infty },\\ \underset{n\to \mathrm{\infty }}{lim sup}{x}_n& =-\mathrm{\infty }& & \text{ implies }\underset{n\to \mathrm{\infty }}{lim}{x}_n=-\mathrm{\infty }.\end{array}}$$

If ${\displaystyle I=\underset{n\to \mathrm{\infty }}{lim inf}{x}_n}$ and ${\displaystyle S=\underset{n\to \mathrm{\infty }}{lim sup}{x}_n}$, then the interval ${\displaystyle [I,S]}$ need not contain any of the numbers ${\displaystyle x_n,}$ but every slight enlargement ${\displaystyle [I-ϵ,S+ϵ],}$ for arbitrarily small ${\displaystyle ϵ>0,}$ will contain ${\displaystyle x_n}$ for all but finitely many indices ${\displaystyle n.}$ In fact, the interval ${\displaystyle [I,S]}$ is the smallest closed interval with this property. We can formalize this property like this: there exist subsequences ${\displaystyle x_{k_n}}$ and ${\displaystyle x_{h_n}}$ of ${\displaystyle x_n}$ (where ${\displaystyle k_n}$ and ${\displaystyle h_n}$ are increasing) for which we have

$${\displaystyle \underset{n\to \mathrm{\infty }}{lim inf}{x}_n+ϵ>x_{h_n}{x}_{k_n}>\underset{n\to \mathrm{\infty }}{lim sup}{x}_n-ϵ}$$

On the other hand, there exists a ${\displaystyle n_0\in \mathbb{N}}$ so that for all ${\displaystyle n\ge n_0}$

$${\displaystyle \underset{n\to \mathrm{\infty }}{lim inf}{x}_n-ϵ<x_n<\underset{n\to \mathrm{\infty }}{lim sup}{x}_n+ϵ}$$

To recapitulate:

• If ${\displaystyle \mathrm{\Lambda }}$ is greater than the limit superior, there are at most finitely many ${\displaystyle x_n}$ greater than ${\displaystyle \mathrm{\Lambda };}$ if it is less, there are infinitely many.
• If ${\displaystyle \lambda }$ is less than the limit inferior, there are at most finitely many ${\displaystyle x_n}$ less than ${\displaystyle \lambda ;}$ if it is greater, there are infinitely many.

Conversely, it can also be shown that:

• If there are infinitely many ${\displaystyle x_n}$ greater than or equal to ${\displaystyle \mathrm{\Lambda }}$, then ${\displaystyle \mathrm{\Lambda }}$ is lesser than or equal to the limit supremum; if there are only finitely many ${\displaystyle x_n}$ greater than ${\displaystyle \mathrm{\Lambda }}$, then ${\displaystyle \mathrm{\Lambda }}$ is greater than or equal to the limit supremum.
• If there are infinitely many ${\displaystyle x_n}$ lesser than or equal to ${\displaystyle \lambda }$, then ${\displaystyle \lambda }$ is greater than or equal to the limit inferior; if there are only finitely many ${\displaystyle x_n}$ lesser than ${\displaystyle \lambda }$, then ${\displaystyle \lambda }$ is lesser than or equal to the limit inferior.

In general,

$${\displaystyle \underset{n}{inf}{x}_n\le \underset{n\to \mathrm{\infty }}{lim inf}{x}_n\le \underset{n\to \mathrm{\infty }}{lim sup}{x}_n\le \underset{n}{sup}{x}_n.}$$

The liminf and limsup of a sequence are respectively the smallest and greatest cluster points.

• For any two sequences of real numbers ${\displaystyle (a_n),(b_n),}$ the limit superior satisfies subadditivity whenever the right side of the inequality is defined (that is, not ${\displaystyle \mathrm{\infty }-\mathrm{\infty }}$ or ${\displaystyle -\mathrm{\infty }+\mathrm{\infty }}$):
  $${\displaystyle \underset{n\to \mathrm{\infty }}{lim sup}(a_n+b_n)\le \underset{n\to \mathrm{\infty }}{lim sup}{a}_n+\underset{n\to \mathrm{\infty }}{lim sup}{b}_n.}$$

  

Analogously, the limit inferior satisfies superadditivity:

$${\displaystyle \underset{n\to \mathrm{\infty }}{lim inf}(a_n+b_n)\ge \underset{n\to \mathrm{\infty }}{lim inf}{a}_n+\underset{n\to \mathrm{\infty }}{lim inf}{b}_n.}$$

In the particular case that one of the sequences actually converges, say ${\displaystyle a_n\to a,}$ then the inequalities above become equalities (with ${\displaystyle \underset{n\to \mathrm{\infty }}{lim sup}{a}_n}$ or ${\displaystyle \underset{n\to \mathrm{\infty }}{lim inf}{a}_n}$ being replaced by ${\displaystyle a}$).

• For any two sequences of non-negative real numbers ${\displaystyle (a_n),(b_n),}$ the inequalities
  $${\displaystyle \underset{n\to \mathrm{\infty }}{lim sup}(a_n{b}_n)\le \left(\underset{n\to \mathrm{\infty }}{lim sup}{a}_n\right)\left(\underset{n\to \mathrm{\infty }}{lim sup}{b}_n\right)}$$

  
  and
  $${\displaystyle \underset{n\to \mathrm{\infty }}{lim inf}(a_n{b}_n)\ge \left(\underset{n\to \mathrm{\infty }}{lim inf}{a}_n\right)\left(\underset{n\to \mathrm{\infty }}{lim inf}{b}_n\right)}$$

  

hold whenever the right-hand side is not of the form ${\displaystyle 0\cdot \mathrm{\infty }.}$

If ${\displaystyle \underset{n\to \mathrm{\infty }}{lim}{a}_n=A}$ exists (including the case ${\displaystyle A=+\mathrm{\infty }}$) and ${\displaystyle B=\underset{n\to \mathrm{\infty }}{lim sup}{b}_n,}$ then ${\displaystyle \underset{n\to \mathrm{\infty }}{lim sup}\left(a_n{b}_n\right)=AB}$ provided that ${\displaystyle AB}$ is not of the form ${\displaystyle 0\cdot \mathrm{\infty }.}$

Examples

• As an example, consider the sequence given by the sine function: ${\displaystyle x_n=\sin (n).}$ Using the fact that π is irrational, it follows that
  $${\displaystyle \underset{n\to \mathrm{\infty }}{lim inf}{x}_n=-1}$$

  
  and
  $${\displaystyle \underset{n\to \mathrm{\infty }}{lim sup}{x}_n=+1.}$$

  
  (This is because the sequence ${\displaystyle \{1,2,3,\dots \}}$ is equidistributed mod 2π, a consequence of the equidistribution theorem.)

• An example from number theory is
  $${\displaystyle \underset{n\to \mathrm{\infty }}{lim inf}(p_{n+1}-p_n),}$$

  
  where ${\displaystyle p_n}$ is the ${\displaystyle n}$-th prime number.

The value of this limit inferior is conjectured to be 2 – this is the twin prime conjecture – but as of April 2014 has only been proven to be less than or equal to 246. The corresponding limit superior is ${\displaystyle +\mathrm{\infty }}$, because there are arbitrarily large gaps between consecutive primes.

Real-valued functions

Assume that a function is defined from a subset of the real numbers to the real numbers. As in the case for sequences, the limit inferior and limit superior are always well-defined if we allow the values +∞ and −∞; in fact, if both agree then the limit exists and is equal to their common value (again possibly including the infinities). For example, given ${\displaystyle f(x)=\sin (1/x)}$, we have ${\displaystyle \underset{x\to 0}{lim sup}f(x)=1}$ and ${\displaystyle \underset{x\to 0}{lim inf}f(x)=-1}$. The difference between the two is a rough measure of how "wildly" the function oscillates, and in observation of this fact, it is called the oscillation of f at 0. This idea of oscillation is sufficient to, for example, characterize Riemann-integrable functions as continuous except on a set of measure zero. Note that points of nonzero oscillation (i.e., points at which f is "badly behaved") are discontinuities which, unless they make up a set of zero, are confined to a negligible set.

Functions from topological spaces to complete lattices

Functions from metric spaces

There is a notion of limsup and liminf for functions defined on a metric space whose relationship to limits of real-valued functions mirrors that of the relation between the limsup, liminf, and the limit of a real sequence. Take a metric space ${\displaystyle X}$, a subspace ${\displaystyle E}$ contained in ${\displaystyle X}$, and a function ${\displaystyle f:E\to \mathbb{R}}$. Define, for any limit point ${\displaystyle a}$ of ${\displaystyle E}$,

${\displaystyle \underset{x\to a}{lim sup}f(x)=\underset{\epsilon \to 0}{lim}\left(sup\{f(x):x\in E\cap B(a,\epsilon )\setminus \{a\}\}\right)}$

and

${\displaystyle \underset{x\to a}{lim inf}f(x)=\underset{\epsilon \to 0}{lim}\left(inf\{f(x):x\in E\cap B(a,\epsilon )\setminus \{a\}\}\right)}$

where ${\displaystyle B(a,\epsilon )}$ denotes the metric ball of radius ${\displaystyle \epsilon }$ about ${\displaystyle a}$.

Note that as ε shrinks, the supremum of the function over the ball is monotone decreasing, so we have

${\displaystyle \underset{x\to a}{lim sup}f(x)=\underset{\epsilon >0}{inf}\left(sup\{f(x):x\in E\cap B(a,\epsilon )\setminus \{a\}\}\right)}$

and similarly

${\displaystyle \underset{x\to a}{lim inf}f(x)=\underset{\epsilon >0}{sup}\left(inf\{f(x):x\in E\cap B(a,\epsilon )\setminus \{a\}\}\right).}$

Functions from topological spaces

This finally motivates the definitions for general topological spaces. Take X, E and a as before, but now let X be a topological space. In this case, we replace metric balls with neighborhoods:

${\displaystyle \underset{x\to a}{lim sup}f(x)=inf\{sup\{f(x):x\in E\cap U\setminus \{a\}\}:U\mathrm{o}\mathrm{p}\mathrm{e}\mathrm{n},a\in U,E\cap U\setminus \{a\}\ne \mathrm{\varnothing }\}}$

${\displaystyle \underset{x\to a}{lim inf}f(x)=sup\{inf\{f(x):x\in E\cap U\setminus \{a\}\}:U\mathrm{o}\mathrm{p}\mathrm{e}\mathrm{n},a\in U,E\cap U\setminus \{a\}\ne \mathrm{\varnothing }\}}$

(there is a way to write the formula using "lim" using nets and the neighborhood filter). This version is often useful in discussions of semi-continuity which crop up in analysis quite often. An interesting note is that this version subsumes the sequential version by considering sequences as functions from the natural numbers as a topological subspace of the extended real line, into the space (the closure of N in [−∞,∞], the extended real number line, is N ∪ {∞}.)

Sequences of sets

The power set ℘(X) of a set X is a complete lattice that is ordered by set inclusion, and so the supremum and infimum of any set of subsets (in terms of set inclusion) always exist. In particular, every subset Y of X is bounded above by X and below by the empty set ∅ because ∅ ⊆ Y ⊆ X. Hence, it is possible (and sometimes useful) to consider superior and inferior limits of sequences in ℘(X) (i.e., sequences of subsets of X).

There are two common ways to define the limit of sequences of sets. In both cases:

• The sequence accumulates around sets of points rather than single points themselves. That is, because each element of the sequence is itself a set, there exist accumulation sets that are somehow nearby to infinitely many elements of the sequence.
• The supremum/superior/outer limit is a set that joins these accumulation sets together. That is, it is the union of all of the accumulation sets. When ordering by set inclusion, the supremum limit is the least upper bound on the set of accumulation points because it contains each of them. Hence, it is the supremum of the limit points.
• The infimum/inferior/inner limit is a set where all of these accumulation sets meet. That is, it is the intersection of all of the accumulation sets. When ordering by set inclusion, the infimum limit is the greatest lower bound on the set of accumulation points because it is contained in each of them. Hence, it is the infimum of the limit points.
• Because ordering is by set inclusion, then the outer limit will always contain the inner limit (i.e., lim inf X_n ⊆ lim sup X_n). Hence, when considering the convergence of a sequence of sets, it generally suffices to consider the convergence of the outer limit of that sequence.

The difference between the two definitions involves how the topology (i.e., how to quantify separation) is defined. In fact, the second definition is identical to the first when the discrete metric is used to induce the topology on X.

General set convergence

See also: Kuratowski convergence and Subsequential limit

A sequence of sets in a metrizable space ${\displaystyle X}$ approaches a limiting set when the elements of each member of the sequence approach the elements of the limiting set. In particular, if ${\displaystyle (X_n)}$ is a sequence of subsets of ${\displaystyle X,}$ then:

• ${\displaystyle lim sup{X}_n,}$ which is also called the outer limit, consists of those elements which are limits of points in ${\displaystyle X_n}$ taken from (countably) infinitely many ${\displaystyle n.}$ That is, ${\displaystyle x\in lim sup{X}_n}$ if and only if there exists a sequence of points ${\displaystyle (x_k)}$ and a subsequence ${\displaystyle (X_{n_k})}$ of ${\displaystyle (X_n)}$ such that ${\displaystyle x_k\in X_{n_k}}$ and ${\displaystyle \underset{k\to \mathrm{\infty }}{lim}{x}_k=x.}$
• ${\displaystyle lim inf{X}_n,}$ which is also called the inner limit, consists of those elements which are limits of points in ${\displaystyle X_n}$ for all but finitely many ${\displaystyle n}$ (that is, cofinitely many ${\displaystyle n}$). That is, ${\displaystyle x\in lim inf{X}_n}$ if and only if there exists a sequence of points ${\displaystyle (x_k)}$ such that ${\displaystyle x_k\in X_k}$ and ${\displaystyle \underset{k\to \mathrm{\infty }}{lim}{x}_k=x.}$

The limit ${\displaystyle lim{X}_n}$ exists if and only if ${\displaystyle lim inf{X}_n}$ and ${\displaystyle lim sup{X}_n}$ agree, in which case ${\displaystyle lim{X}_n=lim sup{X}_n=lim inf{X}_n.}$ The outer and inner limits should not be confused with the set-theoretic limits superior and inferior, as the latter sets are not sensitive to the topological structure of the space.

Special case: discrete metric

This is the definition used in measure theory and probability. Further discussion and examples from the set-theoretic point of view, as opposed to the topological point of view discussed below, are at set-theoretic limit.

By this definition, a sequence of sets approaches a limiting set when the limiting set includes elements which are in all except finitely many sets of the sequence and does not include elements which are in all except finitely many complements of sets of the sequence. That is, this case specializes the general definition when the topology on set X is induced from the discrete metric.

Specifically, for points x, y ∈ X, the discrete metric is defined by

${\displaystyle d(x,y):=\{\begin{array}{ll}0& \text{if }x=y,\\ 1& \text{if }x\ne y,\end{array}}$

under which a sequence of points (x_k) converges to point x ∈ X if and only if x_k = x for all but finitely many k. Therefore, if the limit set exists it contains the points and only the points which are in all except finitely many of the sets of the sequence. Since convergence in the discrete metric is the strictest form of convergence (i.e., requires the most), this definition of a limit set is the strictest possible.

If (X_n) is a sequence of subsets of X, then the following always exist:

• lim sup X_n consists of elements of X which belong to X_n for infinitely many n (see countably infinite). That is, x ∈ lim sup X_n if and only if there exists a subsequence (X_nk) of (X_n) such that x ∈ X_nk for all k.
• lim inf X_n consists of elements of X which belong to X_n for all except finitely many n (i.e., for cofinitely many n). That is, x ∈ lim inf X_n if and only if there exists some m > 0 such that x ∈ X_n for all n > m.

Observe that x ∈ lim sup X_n if and only if x ∉ lim inf X_n^c.

• lim X_n exists if and only if lim inf X_n and lim sup X_n agree, in which case lim X_n = lim sup X_n = lim inf X_n.

In this sense, the sequence has a limit so long as every point in X either appears in all except finitely many X_n or appears in all except finitely many X_n^c. 

Using the standard parlance of set theory, set inclusion provides a partial ordering on the collection of all subsets of X that allows set intersection to generate a greatest lower bound and set union to generate a least upper bound. Thus, the infimum or meet of a collection of subsets is the greatest lower bound while the supremum or join is the least upper bound. In this context, the inner limit, lim inf X_n, is the largest meeting of tails of the sequence, and the outer limit, lim sup X_n, is the smallest joining of tails of the sequence. The following makes this precise.

• Let I_n be the meet of the n^th tail of the sequence. That is,

${\displaystyle \begin{array}{rl}{I}_n& =inf\{X_m:m\in \{n,n+1,n+2,\dots \}\}\\ & =\bigcap _{m=n}^{\mathrm{\infty }}{X}_m=X_n\cap X_{n+1}\cap X_{n+2}\cap \cdots .\end{array}}$

The sequence (I_n) is non-decreasing (i.e. I_n ⊆ I_n+1) because each I_n+1 is the intersection of fewer sets than I_n. The least upper bound on this sequence of meets of tails is

${\displaystyle \begin{array}{rl}\underset{n\to \mathrm{\infty }}{lim inf}{X}_n& =sup\{inf\{X_m:m\in \{n,n+1,\dots \}\}:n\in \{1,2,\dots \}\}\\ & =\bigcup _{n=1}^{\mathrm{\infty }}\left(\bigcap _{m=n}^{\mathrm{\infty }}{X}_m\right).\end{array}}$

So the limit infimum contains all subsets which are lower bounds for all but finitely many sets of the sequence.

• Similarly, let J_n be the join of the n^th tail of the sequence. That is,

${\displaystyle \begin{array}{rl}{J}_n& =sup\{X_m:m\in \{n,n+1,n+2,\dots \}\}\\ & =\bigcup _{m=n}^{\mathrm{\infty }}{X}_m=X_n\cup X_{n+1}\cup X_{n+2}\cup \cdots .\end{array}}$

The sequence (J_n) is non-increasing (i.e. J_n ⊇ J_n+1) because each J_n+1 is the union of fewer sets than J_n. The greatest lower bound on this sequence of joins of tails is

${\displaystyle \begin{array}{rl}\underset{n\to \mathrm{\infty }}{lim sup}{X}_n& =inf\{sup\{X_m:m\in \{n,n+1,\dots \}\}:n\in \{1,2,\dots \}\}\\ & =\bigcap _{n=1}^{\mathrm{\infty }}\left(\bigcup _{m=n}^{\mathrm{\infty }}{X}_m\right).\end{array}}$

So the limit supremum is contained in all subsets which are upper bounds for all but finitely many sets of the sequence.

Examples

The following are several set convergence examples. They have been broken into sections with respect to the metric used to induce the topology on set X.

Using the discrete metric

• The Borel–Cantelli lemma is an example application of these constructs.

Using either the discrete metric or the Euclidean metric

• Consider the set X = {0,1} and the sequence of subsets:

${\displaystyle (X_n)=(\{0\},\{1\},\{0\},\{1\},\{0\},\{1\},\dots ).}$

The "odd" and "even" elements of this sequence form two subsequences, ({0}, {0}, {0}, ...) and ({1}, {1}, {1}, ...), which have limit points 0 and 1, respectively, and so the outer or superior limit is the set {0,1} of these two points. However, there are no limit points that can be taken from the (X_n) sequence as a whole, and so the interior or inferior limit is the empty set { }. That is,

• lim sup X_n = {0,1}
• lim inf X_n = { }

However, for (Y_n) = ({0}, {0}, {0}, ...) and (Z_n) = ({1}, {1}, {1}, ...):

• lim sup Y_n = lim inf Y_n = lim Y_n = {0}
• lim sup Z_n = lim inf Z_n = lim Z_n = {1}

• Consider the set X = {50, 20, −100, −25, 0, 1} and the sequence of subsets:

${\displaystyle (X_n)=(\{50\},\{20\},\{-100\},\{-25\},\{0\},\{1\},\{0\},\{1\},\{0\},\{1\},\dots ).}$

As in the previous two examples,

• lim sup X_n = {0,1}
• lim inf X_n = { }

That is, the four elements that do not match the pattern do not affect the lim inf and lim sup because there are only finitely many of them. In fact, these elements could be placed anywhere in the sequence. So long as the tails of the sequence are maintained, the outer and inner limits will be unchanged. The related concepts of essential inner and outer limits, which use the essential supremum and essential infimum, provide an important modification that "squashes" countably many (rather than just finitely many) interstitial additions.

Using the Euclidean metric

• Consider the sequence of subsets of rational numbers:

${\displaystyle (X_n)=(\{0\},\{1\},\{1/2\},\{1/2\},\{2/3\},\{1/3\},\{3/4\},\{1/4\},\dots ).}$

The "odd" and "even" elements of this sequence form two subsequences, ({0}, {1/2}, {2/3}, {3/4}, ...) and ({1}, {1/2}, {1/3}, {1/4}, ...), which have limit points 1 and 0, respectively, and so the outer or superior limit is the set {0,1} of these two points. However, there are no limit points that can be taken from the (X_n) sequence as a whole, and so the interior or inferior limit is the empty set { }. So, as in the previous example,

• lim sup X_n = {0,1}
• lim inf X_n = { }

However, for (Y_n) = ({0}, {1/2}, {2/3}, {3/4}, ...) and (Z_n) = ({1}, {1/2}, {1/3}, {1/4}, ...):

• lim sup Y_n = lim inf Y_n = lim Y_n = {1}
• lim sup Z_n = lim inf Z_n = lim Z_n = {0}

In each of these four cases, the elements of the limiting sets are not elements of any of the sets from the original sequence.

• The Ω limit (i.e., limit set) of a solution to a dynamic system is the outer limit of solution trajectories of the system. Because trajectories become closer and closer to this limit set, the tails of these trajectories converge to the limit set.

• For example, an LTI system that is the cascade connection of several stable systems with an undamped second-order LTI system (i.e., zero damping ratio) will oscillate endlessly after being perturbed (e.g., an ideal bell after being struck). Hence, if the position and velocity of this system are plotted against each other, trajectories will approach a circle in the state space. This circle, which is the Ω limit set of the system, is the outer limit of solution trajectories of the system. The circle represents the locus of a trajectory corresponding to a pure sinusoidal tone output; that is, the system output approaches/approximates a pure tone.

Generalized definitions

The above definitions are inadequate for many technical applications. In fact, the definitions above are specializations of the following definitions.

Definition for a set

The limit inferior of a set X ⊆ Y is the infimum of all of the limit points of the set. That is,

${\displaystyle lim infX:=inf\{x\in Y:x\text{ is a limit point of }X\}}$

Similarly, the limit superior of X is the supremum of all of the limit points of the set. That is,

${\displaystyle lim supX:=sup\{x\in Y:x\text{ is a limit point of }X\}}$

Note that the set X needs to be defined as a subset of a partially ordered set Y that is also a topological space in order for these definitions to make sense. Moreover, it has to be a complete lattice so that the suprema and infima always exist. In that case every set has a limit superior and a limit inferior. Also note that the limit inferior and the limit superior of a set do not have to be elements of the set.

Definition for filter bases

See also: Filters in topology

Take a topological space X and a filter base B in that space. The set of all cluster points for that filter base is given by

${\displaystyle \bigcap \{{\overline{B}}_0:B_0\in B\}}$

where ${\displaystyle {\overline{B}}_0}$ is the closure of ${\displaystyle B_0}$. This is clearly a closed set and is similar to the set of limit points of a set. Assume that X is also a partially ordered set. The limit superior of the filter base B is defined as

${\displaystyle lim supB:=sup\bigcap \{{\overline{B}}_0:B_0\in B\}}$

when that supremum exists. When X has a total order, is a complete lattice and has the order topology,

${\displaystyle lim supB=inf\{sup{B}_0:B_0\in B\}.}$

Similarly, the limit inferior of the filter base B is defined as

${\displaystyle lim infB:=inf\bigcap \{{\overline{B}}_0:B_0\in B\}}$

when that infimum exists; if X is totally ordered, is a complete lattice, and has the order topology, then

${\displaystyle lim infB=sup\{inf{B}_0:B_0\in B\}.}$

If the limit inferior and limit superior agree, then there must be exactly one cluster point and the limit of the filter base is equal to this unique cluster point.

Specialization for sequences and nets

Note that filter bases are generalizations of nets, which are generalizations of sequences. Therefore, these definitions give the limit inferior and limit superior of any net (and thus any sequence) as well. For example, take topological space ${\displaystyle X}$ and the net ${\displaystyle (x_{\alpha }{)}_{\alpha \in A}}$, where ${\displaystyle (A,\le )}$ is a directed set and ${\displaystyle x_{\alpha }\in X}$ for all ${\displaystyle \alpha \in A}$. The filter base ("of tails") generated by this net is ${\displaystyle B}$ defined by

${\displaystyle B:=\{\{x_{\alpha }:{\alpha }_0\le \alpha \}:{\alpha }_0\in A\}.}$

Therefore, the limit inferior and limit superior of the net are equal to the limit superior and limit inferior of ${\displaystyle B}$ respectively. Similarly, for topological space ${\displaystyle X}$, take the sequence ${\displaystyle (x_n)}$ where ${\displaystyle x_n\in X}$ for any ${\displaystyle n\in \mathbb{N}}$. The filter base ("of tails") generated by this sequence is ${\displaystyle C}$ defined by

${\displaystyle C:=\{\{x_n:n_0\le n\}:n_0\in \mathbb{N}\}.}$

Therefore, the limit inferior and limit superior of the sequence are equal to the limit superior and limit inferior of ${\displaystyle C}$ respectively.

Critique of the Kantian philosophy

From Wikipedia, the free encyclopedia
https://en.wikipedia.org/wiki/Critique_of_the_Kantian_philosophy

"Critique of the Kantian philosophy" (German: "Kritik der Kantischen Philosophie") is a criticism Arthur Schopenhauer appended to the first volume of his The World as Will and Representation (1818). He wanted to show Immanuel Kant's errors so that Kant's merits would be appreciated and his achievements furthered.

At the time he wrote his criticism, Schopenhauer was acquainted only with the second (1787) edition of Kant's Critique of Pure Reason. When he later read the first (1781) edition, he said that many of Kant's contradictions were not evident.

Kant's merits

According to Schopenhauer's essay, Kant's three main merits are as follows:

1. The distinction of the phenomenon from the thing-in-itself (Ding an sich)
   • The intellect mediates between things and knowledge
   • Locke's primary qualities result from the mind's activity, just as his secondary qualities result from receptivity at any of the five senses
   • A priori knowledge is separate from a posteriori knowledge
   • The ideal and the real are diverse from each other
   • Transcendental philosophy goes beyond dogmatic philosophy's "eternal truths", such as the principle of contradiction and the principle of sufficient reason. It shows that those "truths" are based on necessary forms of thought that exist in the mind.
2. The explanation of how the moral significance of human conduct is different from the laws that are concerned with phenomena
   • The significance is directly related to the thing-in-itself, the innermost nature of the world
3. Religious scholastic philosophy is completely overthrown by the demonstration of the impossibility of proofs for speculative theology and also for rational psychology, or reasoned study of the soul

Schopenhauer also said that Kant's discussion, on pages A534 to A550, of the contrast between empirical and intelligible characters is one of Kant's most profound ideas. Schopenhauer asserted that it is among the most admirable things ever said by a human.

• The empirical character of a phenomenon is completely determined
• The intelligible character of a phenomenon is free. It is the thing-in-itself which is experienced as a phenomenon.

Kant's faults

Fundamental error

Perceptions and concepts

Kant wanted to make the table of judgments the key to all knowledge. In so doing, he was concerned with making a system and did not think of defining terms such as perception and conception, as well as reason, understanding, subject, object, and others.

Fundamental error: Kant did not distinguish between the concrete, intuitive, perceptual knowledge of objects and the abstract, discursive, conceptual, knowledge of thoughts.

• Kant began his investigation into knowledge of perceived objects by considering indirect, reflective knowledge of concepts instead of direct, intuitive knowledge of perceptions.
• For Kant, there is absolutely no knowledge of an object unless there is thought which employs abstract concepts. For him, perception is not knowledge because it is not thought. In general, Kant claimed that perception is mere sensation.
  • In accordance with Kant's claim, non-human animals would not be able to know objects. Animals would only know impressions on their sense organs, which Kant mistakenly called perception. Kant had erroneously asserted that full, perceived objects, not mere sensations, were given to the mind by the sense organs. Perception, however, according to Schopenhauer, is intellectual and is a product of the Understanding. Perception of an object does not result from the mere data of the senses. It requires the Understanding. Therefore, if animals do not have Understanding, in accordance with Kant, then they have only Sensation, which, Schopenhauer claimed, gives only raw sense data, not perceived objects.
• Schopenhauer considered the following sentences on page A253 of the Critique of Pure Reason to encapsulate all of Kant's errors:
  • If all thought (by means of categories) is taken away from empirical knowledge, no knowledge of any object remains, because nothing can be thought by mere intuition or perception. The simple fact that there is within me an affection of my sensibility, establishes in no way any relation of such a representation to any object.
  • On page A253, Kant stated that no knowledge of any object would remain if all thought by means of categories were removed from empirical knowledge.
    • Schopenhauer claimed that perception occurs without conceptual thought.
  • On page A253, Kant stated that a concept without an intuition is not empty. It still has the form of thought.
    • Schopenhauer claimed that perceived representations are the content of a concept. Without them, the concept is empty.

Secondary errors

Transcendental analytic

• Kant asserted that metaphysics is knowledge a priori, or before experience. As a result, he concluded that the source of metaphysics cannot be inner or outer experience.
  • Schopenhauer claimed that metaphysics must understand inner and outer experience in order to know the world and not empty forms. Kant did not prove that the material for knowing the world is outside of the experience of the world and merely in the forms of knowledge.
• Kant's writing was obscure.
• Kant took the Greek word noumena, which meant "that which is thought," and used it to mean "things-in-themselves." (See Sextus Empiricus, Outlines of Pyrrhonism, Book I, Chapter 13: "Anaxagoras opposed what is thought (noumena) to what appears or is perceived (phenomena).")
• Kant tried to create a logical, overly-symmetrical system without reflecting on its contents.
  • He didn't clearly explain the meaning of and relationships between represented objects, representing subjects, existence, truth, illusion, error, sensations, judgments, words, concepts, perceptions, understanding, and reason.

Concepts

• • Kant didn't clearly explain concepts in general:
    • Concepts of the understanding (common concepts and categories).
    • Concepts of Reason (Ideas of God, Freedom, and Immortality).
  • He divided reason into theoretical and practical, making practical reason the source of virtuous conduct.

Idealism

• Kant altered his first edition to:
  • suppress the idealistic assertion that objects are conditioned by the knowing subject;

Object-in-itself and thing-in-itself

According to Schopenhauer, there is a difference between an object-in-itself and a thing-in-itself. There is no object-in-itself. An object is always an object for a subject. An object is really a representation of an object. On the other hand, a thing-in-itself, for Kant, is completely unknown. It cannot be spoken of at all without employing categories (pure concepts of the understanding). A thing-in-itself is that which appears to an observer when the observer experiences a representation.

• Kant altered his first edition to:
  • claim that the spatially external thing-in-itself causes sensations in the sense organs of the knowing subject.
• Kant tried to explain how:
  • a perceived object, not mere raw sensation, is given to the mind by sensibility (sensation, space, and time), and
  • how the human understanding produces an experienced object by thinking twelve categories.
• Kant doesn't explain how something external causes sensation in a sense organ.
• He didn't explain whether the object of experience (the object of knowledge which is the result of the application of the categories) is a perceptual representation or an abstract concept. He mixed up the perceptible and the abstract so that an absurd hybrid of the two resulted.
  • There is a contradiction between the object experienced by the senses and the object experienced by the understanding.
    • Kant claims that representation of an object occurs both
      • through reception of one or more of the five senses, and
      • through the activity of the understanding's twelve categories.
    • Sensation and understanding are separate and distinct abilities. Yet, for Kant, an object is known through each of them.
    • This contradiction is the source of the obscurity of the Transcendental Logic.
  • Kant's incorrect triple distinction:
    • Representation (given to one or more of the 5 senses, and to the sensibilities of space and time)
    • Object that is represented (thought through the 12 categories)
    • Thing-in-itself (cannot be known).
  • Schopenhauer claimed that Kant's represented object is false. The true distinction is only between the representation and the thing-in-itself.
  • For Schopenhauer, the law of causality, which relates only to the representation and not to the thing-in-itself, is the real and only form of the understanding. The other 11 categories are therefore unnecessary because there is no represented object to be thought through them.
  • Kant sometimes spoke of the thing-in-itself as though it was an object that caused changes in a subject's senses. Schopenhauer affirmed that the thing-in-itself was totally different from phenomena and therefore had nothing to do with causality or being an object for a subject.
• Excessive fondness for symmetry:
  • Origin of Kant's Transcendental Logic:
    • As pure intuitions (in the Transcendental Aesthetic) were the basis of empirical intuitions,
    • pure concepts (in the Transcendental Logic) were made the basis of empirical concepts.
    • As the Transcendental Aesthetic was the a priori basis of mathematics,
    • the Transcendental Logic was made the a priori basis of logic.
• After discovering empirical perception is based on two forms of a priori perception (space and time), Kant tried to demonstrate that empirical knowledge is based on an analogous a priori knowledge (categories).

Schemata

• He went too far when he claimed that the schemata of the pure concepts of the understanding (the categories) are analogous to a schema of empirically acquired concepts.
  • A schema of empirical perception is a sketchy, imagined perception. Thus, a schema is the mere imagined form or outline, so to speak, of a real perception. It is related to an empirical abstract concept to show that the concept is not mere word-play but has indeed been based on real perceptions. These perceptions are the actual, material content of the empirical abstract concept.
  • A schema of pure concepts is supposed to be a pure perception. There is supposed to be a schema for each of the pure concepts (categories). Kant overlooked the fact that these pure concepts, being pure, have no perceptual content. They gain this content from empirical perception. Kant's schemata of pure concepts are entirely undemonstrable and are a merely arbitrary assumption.
  • This demonstrates Kant's purposeful intention to find a pure, a priori analogical basis for every empirical, a posteriori mental activity.

Judgments/categories

• Derived all philosophical knowledge from the table of judgments.
• Made the table of categories the basis for every assertion about the physical and the metaphysical.
  • Derived pure concepts of the understanding (categories) from reason. But the Transcendental Analytic was supposed to reference only the sensibility of the sense organs and also the mind's way of understanding objects. It was not supposed to be concerned with reason.
    • Categories of quantity were based on judgments of quantity. But these judgments relate to reason, not understanding. They involve logical inclusion or exclusion of concepts with each other, as follows:
      • Universal judgment: All A are x; Particular judgment: Some A are x; Singular judgment: This one A is x.
      • Note: The word "quantity" was poorly chosen to designate mutual relations between abstract concepts.
    • Categories of quality were based on judgments of quality. But these judgments also are related only to reason, not to understanding. Affirmation and denial are relations between concepts in a verbal judgment. They have nothing to do with perceptual reality for the understanding. Kant also included infinite judgments, but only for the sake of architectonic symmetry. They have no meaning in Kant's context.
      • The term "quality" was chosen because it has usually been opposed to "quantity." But here it means only affirmation and denial in a judgment.
  • The categorical relation (A is x) is simply the general connection of a subject concept with a predicate concept in a statement. It includes the hypothetical and disjunctive sub-relations. It also includes the judgments of quality (affirmation, negation) and judgments of quantity (inclusional relationships between concepts). Kant made separate categories from these sub-relations. He used indirect, abstract knowledge to analyze direct, perceptual knowledge.
    • Our certain knowledge of the physical persistence of substance, or the conservation of matter, is derived, by Kant, from the category of subsistence and inherence. But this is merely based on the connection of a linguistic subject with its predicate.
  • With judgments of relation, the hypothetical judgment (if A, then B) does not correspond only to the law of causality. This judgment is also associated with three other roots of the principle of sufficient reason. Abstract reasoning does not disclose the distinction between these four kinds of ground. Knowledge from perception is required.
    • reason of knowing (logical inference);
    • reason of acting (law of motivation);
    • reason of being (spatial and temporal relations, including the arithmetical sequences of numbers and the geometrical positions of points, lines, and surfaces).
  • Disjunctive judgments derive from the logical law of thought of the excluded middle (A is either A or not-A). This relates to reason, not to the understanding. For the purpose of symmetry, Kant asserted that the physical analog of this logical law was the category of community or reciprocal effect. However, it is the opposite, since the logical law refers to mutually exclusive predicates, not inclusive.
    • Schopenhauer asserted that there is no reciprocal effect. It is only a superfluous synonym for causality. For architectonic symmetry, Kant created a separate a priori function in the understanding for reciprocal effect. Actually, there is only an alternating succession of states, a chain of causes and effects.
  • Modal categories of possible, actual, and necessary are not special, original cognized forms. They are derived from the principle of sufficient reason (ground).
    • Possibility is a general, mental abstraction. It refers to abstract concepts, which are solely related to the ability to reason or logically infer.
    • There is no difference between actuality (existence) and necessity.
    • Necessity is a consequence from a given ground (reason).

Transcendental dialectic

Reason

• Kant defined reason as the faculty or power of principles. He claimed that principles provide us with synthetical knowledge from mere concepts (A 301; B 358). However, knowledge from mere concepts, without perception, is analytical, not synthetical. Synthetical knowledge requires the combination of two concepts, plus a third thing. This third thing is pure intuition or perception, if it is a priori, and empirical perception, if it is a posteriori.
• According to Kant's principle of reason, everything that is conditioned is part of a total series of conditions. The essential nature of reason tries to find something unconditioned that functions as a beginning of the series.
  • But Schopenhauer claimed that the demand is only for a sufficient reason or ground. It extends merely to the completeness of the determinations of the nearest or next cause, not to an absolute first cause.
• Kant claimed that everyone's reason leads them to assume three unconditioned absolutes. These are God, the soul, and the total world. The unconditioned absolutes are symmetrically derived by Kant from three kinds of syllogism as the result of three categories of relation.
  • Schopenhauer stated that the soul and the total world are not unconditioned because they are supposed, by believers, to be conditioned by God.
  • Schopenhauer also stated that everyone's reason does not lead to these three unconditioned absolutes. Buddhists are nontheists. Only Judaism and its derivatives, Christianity and Islam, are monotheistic. Exhaustive and extensive historical research would be needed to validate Kant's claim about the universality of reason's three unconditioned absolutes.

Ideas of reason

• Kant called God, soul, and total world (cosmos) Ideas of Reason. In doing so, he appropriated Plato's word "Idea" and ambiguously changed its settled meaning. Plato's Ideas are models or standards from which copies are generated. The copies are visible objects of perception. Kant's Ideas of Reason are not accessible to knowledge of perception. They are barely understandable through abstract knowledge of concepts.
• Fondness for symmetry led Kant to derive, as necessary, the concept of the soul from the paralogisms of rational psychology. He did so by applying the demand for the unconditional to the concept of substance, which is the first category of relation.
• Kant claimed that the concept of the soul arose from the concept of the final, unconditioned subject of all predicates of a thing. This was taken from the logical form of the categorical syllogism.
  • Schopenhauer asserted that subjects and predicates are logical. They are concerned only with the relation of abstract concepts in a judgment. They are not concerned with a substance, such as a soul, that contains no material basis.
• The Idea of the total world, cosmos, or universe was said, by Kant, to originate from the hypothetical syllogism (If A is x, then B is y; A is x; Therefore, B is y).
  • Schopenhauer said that all three Ideas (God, soul, and universe) might be derived from the hypothetical syllogism. This is because all of these Ideas are concerned with the dependence of one object on another. When no more dependencies can be imagined, then the unconditioned has been reached.
• Relating the Cosmological Ideas to the Table of Categories
  • Kant stated that the cosmological Ideas, with regard to the limits of the world in time and space, are determined through the category of quantity.
    • Schopenhauer asserted that those Ideas are not related to that category. Quantity is only concerned with the mutual inclusion of exclusion of concepts with each other (All A are x; Some A are x; This A is x).
  • Kant said that the divisibility of matter occurred according to the category of quality. But quality is merely the affirmation or negation in a judgment. Schopenhauer wrote that the mechanical divisibility of matter is associated with the quantity of matter, not quality.
    • All of the cosmological ideas should derive from the hypothetical form of syllogism and therefore from the principle of sufficient reason. Kant asserted that divisibility of a whole into ultimate parts was based on the principle of sufficient reason. This is because the ultimate parts are supposed to be the ground conditions and the whole is supposed to be the consequent. However, Schopenhauer claimed that divisibility is instead based on the principle of contradiction. For him, the parts and the whole are actually one. If the ultimate parts are thought away, then the whole is also thought away.
  • According to Schopenhauer, the fourth antinomy is redundant. It is an unnecessary repetition of the third antinomy. This arrangement was formed for the purpose of maintaining the architectonic symmetry of the category table.
    • The thesis of the third antinomy asserts the existence of the causality of freedom. This is the same as the primary cause of the world.
    • The thesis of the fourth antinomy asserts the existence of an absolutely necessary Being that is the cause of the world. Kant associated this with modality because through the first cause, the contingent becomes necessary.
• Schopenhauer calls the whole antinomy of cosmology a mere sham fight. He said that Kant only pretended that there is a necessary antinomy in reason.
  • In all four antinomies, the proof of the thesis is a sophism.
  • The proof of each antithesis, however, is an inevitable conclusion from premisses that are derived from the absolutely certain laws of the phenomenal world.
• The theses are sophisms, according to Schopenhauer.
  • First Cosmological Antinomy's Thesis:
    • Purports to discuss beginning of time but instead discusses end or completion of series of times.
    • Arbitrarily presupposes that the world is given as a whole and is therefore limited.
  • Second Cosmological Antinomy's Thesis:
    • Begs the question by presupposing that a compound is an accumulation of simple parts.
    • Arbitrarily assumes that all matter is compound instead of an infinitely divisible total.
  • Third Cosmological Antinomy's Thesis:
    • Kant appeals to his principle of pure reason (reason seeks the unconditioned in a series) in order to support causality through freedom. But, according to Schopenhauer, reason seeks the latest, most recent, sufficient cause. It does not seek the most remote first cause.
    • Kant said that the practical concept of freedom is based on the transcendent Idea of freedom, which is an unconditioned cause. Schopenhauer argued that the recognition of freedom comes from the consciousness that the inner essence or thing-in-itself is free will.
  • Fourth Cosmological Antinomy's Thesis:
    • The fourth antinomy is a redundant repetition of the third antinomy. Every conditioned does not presuppose a complete series of conditions which ends with the unconditioned. Instead, every conditioned presupposes only its most recent condition.
• As a solution to the cosmological antinomy, Kant stated:
  • Both sides assumed that the world exists in itself. Therefore, both sides are wrong in the first and second antinomies.
  • Both sides assumed that reason assumes an unconditioned first cause of a series of conditions. Therefore, both sides are correct in the third and fourth antinomies.
  • Schopenhauer disagreed. He said that the solution was that the antitheses are correct in all four antinomies.
• Kant stated that the Transcendental Ideal is a necessary idea of human reason. It is the most real, perfect, powerful entity.
  • Schopenhauer disagreed. He said that his own reason found this idea to be impossible. He was unable to think of any definite object that corresponds to the description.
• The three main objects of scholastic philosophy were the soul, the world, and God. Kant tried to show how they were taken from the three possible major premisses of syllogisms.
  • The soul was derived from the categorical judgment (A is x) and the world was taken from the hypothetical judgment (If A is x, then B is y).
  • For architectonic symmetry, God had to be derived from the remaining disjunctive judgment (A is either x or not-x).
    • Schopenhauer said that the antique philosophers did not mention this derivation, so it can't be necessary to all human reason. Their gods were limited. World-creating gods merely gave form to pre-existing matter. Reason, according to ancient philosophers, did not obtain an idea of a perfect God or Ideal from the disjunctive syllogism.
    • Kant stated that knowledge of particular things results from a continuous process of the limitation of general or universal concepts. The most universal concept would then have contained all reality in itself.
      • According to Schopenhauer, the reverse is true. Knowledge starts from the particular and is extended to the general. General concepts result from abstraction from particulars, retaining only their common element. The most universal concept would thus have the least particular content and be the emptiest.
• Kant alleged that the three transcendent ideas are useful as regulative principles. As such, he claimed, they aid in the advancement of the knowledge of nature.
  • Schopenhauer asserted that Kant was diametrically wrong. The ideas of soul, finite world, and God are hindrances. For example, the search for a simple, immaterial, thinking soul would not be scientifically useful.

Ethics

• Kant claimed that virtue results from practical reason.
  • Schopenhauer claimed that, to the contrary, virtuous conduct has nothing to do with a rational life and may even be opposed to it, as with Machiavellian rational expediency.

Categorical Imperative

• According to Schopenhauer, Kant's Categorical Imperative:
  • Redundantly repeats the ancient command: "don't do to another what you don't want done to you."
  • Is egoistic because its universality includes the person who both gives and obeys the command.
  • Is cold and dead because it is to be followed without love, feeling, or inclination, but merely out of a sense of duty.

Power of judgment

• In the Critique of Pure Reason, Kant claimed that the understanding was the ability to judge. The forms of judgments were said to be the basis of the categories and all philosophy. But in his Critique of Judgment, he called a new, different ability the faculty of judgment. That now resulted in four faculties: sensation, understanding, judging, and reason. Judgment was located between understanding and reason, and contained elements of both.
• Kant's interest in the concept of suitability or expediency resulted in his investigation regarding knowledge of beauty and knowledge of natural purposiveness.

Aesthetics

• As usual, he started from abstract concepts in order to know concrete perceptions. Kant started from the abstract judgment of taste in order to investigate knowledge of beautiful objects of perception.
• Kant was not concerned with beauty itself. His interest was in the question of how a subjective statement or judgment about beauty could be universally valid, as though it concerned an actual quality of an object.

Teleology

• Kant asserted that the subjective statement that nature seems to have been created with a premeditated purpose does not necessarily have objective validity or truth.
• Kant claimed that the apparently purposive, deliberate constitution of organic bodies cannot be explained from merely mechanical causes. ("...it is absurd for man even to entertain any thought ... that maybe another Newton may some day arise to make intelligible to us even the genesis of but a blade of grass from natural laws that no design has ordered [i.e., from mechanical principles].") (Critique of Judgment, §75).
• Schopenhauer said that Kant didn't go far enough. Schopenhauer stated that one province of nature cannot be explained from laws of any other province of nature. He listed examples of separate provinces of nature as being mechanics, chemistry, electricity, magnetism, crystallization, and organics. Kant had only asserted this regarding the organic and the mechanical.

Reactions to Schopenhauer

Paul Guyer

In The Cambridge Companion to Schopenhauer (1999), the philosopher Paul Guyer wrote an article titled "Schopenhauer, Kant, and the Methods of Philosophy." In it, he compared the methods of the two philosophers and in so doing, discussed Schopenhauer's Criticism.

In explaining how objects are experienced, Kant used transcendental arguments. He tried to prove and explain the fundamental principles of knowledge. In so doing, he started by indirectly conceptually reflecting on the conditions that exist in the observing subject that make possible verbal judgments about objective experience.

We shall therefore follow up the pure concepts to their first germs and beginnings in the human understanding...

— A66

In contrast, Schopenhauer's method was to start by a direct examination of perceived objects in experience, not of abstract concepts.

...the solution of the riddle of the world is possible only through the proper connexion of outer with inner experience...

— Appendix p. 428

The fundamental principles of knowledge cannot be transcendentally explained or proved, they can only be immediately, directly known. Such principles are, for example, the permanence of substance, the law of causality, and the mutual interactive relationships between all objects in space. Abstract concepts, for Schopenhauer, are not the starting point of knowledge. They are derived from perceptions, which are the source of all knowledge of the objective world. The world is experienced in two ways: (1.) mental representations that involve space, time, and causality; (2.) our will which is known to control our body.

Guyer stated that Schopenhauer raised important questions regarding the possibility of Kant's transcendental arguments and proofs. However, even though Schopenhauer objected to Kant's method, he accepted many of Kant's conclusions. For example, Kant's description of experience and its relation to space, time, and causality was accepted. Also, the distinction between logical and real relations, as well as the difference between phenomena and things-in-themselves, played an important role in Schopenhauer's philosophy.

In general, the article tries to show how Schopenhauer misunderstood Kant as a result of the disparity between their methods. Where Kant was analyzing the conceptual conditions that resulted in the making of verbal judgments, Schopenhauer was phenomenologically scrutinizing intuitive experience. In one case, though, it is claimed that Schopenhauer raised a very important criticism: his objection to Kant's assertion that a particular event can be known as being successive only if its particular cause is known. Otherwise, almost all of Schopenhauer's criticisms are attributed to his opposite way of philosophizing which starts with the examination of perceptions instead of concepts.

Derek Parfit

In philosopher Derek Parfit's 2011 book On What Matters, Volume 1, Parfit presents an argument against psychological egoism that centers around an apparent equivocation between different senses of the word "want":

The word desire often refers to our sensual desires or appetites, or to our being attracted to something, by finding the thought of it appealing. I shall use desire in a wider sense, which refers to any state of being motivated, or of wanting something to happen and being to some degree disposed to make it happen, if we can. The word want already has both these senses.

Some people think: Whenever people act voluntarily, they are doing what they want to do. Doing what we want is selfish. So everyone always acts selfishly. This argument for Psychological Egoism fails, because it uses the word want first in the wide sense and then in the narrow sense. If I voluntarily gave up my life to save the lives of several strangers, my act would not be selfish, though I would be doing what in the wide sense I wanted to do.

Michael Kelly

Michael Kelly, in the preface to his 1910 book Kant's Ethics and Schopenhauer's Criticism, stated: "Of Kant it may be said that what is good and true in his philosophy would have been buried with him, were it not for Schopenhauer...."

Immanuel Kant

Immanuel Kant himself predicted a response to Schopenhauer's argument that he redundantly repeated the ancient command: "don't do to another what you don't want done to you", that is, the Golden Rule, and famously criticized it for not being sensitive to differences of situation, noting that a prisoner duly convicted of a crime could appeal to the golden rule while asking the judge to release him, pointing out that the judge would not want anyone else to send him to prison, so he should not do so to others. Kant's Categorical Imperative, introduced in Groundwork of the Metaphysic of Morals, is often confused with the Golden Rule. Also, it is exactly for being cold and dead because it is to be followed without love, feeling, or inclination, but merely out of a sense of duty, both in the theory and in its practice, that the Categorical Imperative is absolute, metaphysical and moral.