Hyperreal number

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

Infinitesimals (ε) and infinities (ω) on the hyperreal number line (1/ε = ω/1)

In mathematics, the system of hyperreal numbers is a way of treating infinite and infinitesimal (infinitely small but non-zero) quantities. The hyperreals, or nonstandard reals, *R, are an extension of the real numbers R that contains numbers greater than anything of the form

${\displaystyle 1+1+\cdots +1}$ (for any finite number of terms).

Such numbers are infinite, and their reciprocals are infinitesimals. The term "hyper-real" was introduced by Edwin Hewitt in 1948.

The hyperreal numbers satisfy the transfer principle, a rigorous version of Leibniz's heuristic law of continuity. The transfer principle states that true first-order statements about R are also valid in *R. For example, the commutative law of addition, x + y = y + x, holds for the hyperreals just as it does for the reals; since R is a real closed field, so is *R. Since ${\displaystyle \sin({\pi n})=0}$ for all integers n, one also has ${\displaystyle \sin({\pi H})=0}$ for all hyperintegers H. The transfer principle for ultrapowers is a consequence of Łoś' theorem of 1955.

Concerns about the soundness of arguments involving infinitesimals date back to ancient Greek mathematics, with Archimedes replacing such proofs with ones using other techniques such as the method of exhaustion. In the 1960s, Abraham Robinson proved that the hyperreals were logically consistent if and only if the reals were. This put to rest the fear that any proof involving infinitesimals might be unsound, provided that they were manipulated according to the logical rules that Robinson delineated.

The application of hyperreal numbers and in particular the transfer principle to problems of analysis is called nonstandard analysis. One immediate application is the definition of the basic concepts of analysis such as the derivative and integral in a direct fashion, without passing via logical complications of multiple quantifiers. Thus, the derivative of f(x) becomes ${\displaystyle f'(x)=\operatorname {st} \left({\frac {f(x+\Delta x)-f(x)}{\Delta x}}\right)}$ for an infinitesimal ${\displaystyle \Delta x}$, where st(·) denotes the standard part function, which "rounds off" each finite hyperreal to the nearest real. Similarly, the integral is defined as the standard part of a suitable infinite sum.

The transfer principle

Main article: Transfer principle

The idea of the hyperreal system is to extend the real numbers R to form a system *R that includes infinitesimal and infinite numbers, but without changing any of the elementary axioms of algebra. Any statement of the form "for any number x..." that is true for the reals is also true for the hyperreals. For example, the axiom that states "for any number x, x + 0 = x" still applies. The same is true for quantification over several numbers, e.g., "for any numbers x and y, xy = yx." This ability to carry over statements from the reals to the hyperreals is called the transfer principle. However, statements of the form "for any set of numbers S ..." may not carry over. The only properties that differ between the reals and the hyperreals are those that rely on quantification over sets, or other higher-level structures such as functions and relations, which are typically constructed out of sets. Each real set, function, and relation has its natural hyperreal extension, satisfying the same first-order properties. The kinds of logical sentences that obey this restriction on quantification are referred to as statements in first-order logic.

The transfer principle, however, does not mean that R and *R have identical behavior. For instance, in *R there exists an element ω such that

${\displaystyle 1<\omega ,\quad 1+1<\omega ,\quad 1+1+1<\omega ,\quad 1+1+1+1<\omega ,\ldots .}$

but there is no such number in R. (In other words, *R is not Archimedean.) This is possible because the nonexistence of ω cannot be expressed as a first-order statement.

Use in analysis

Calculus with algebraic functions

Informal notations for non-real quantities have historically appeared in calculus in two contexts: as infinitesimals, like dx, and as the symbol ∞, used, for example, in limits of integration of improper integrals.

As an example of the transfer principle, the statement that for any nonzero number x, 2x ≠ x, is true for the real numbers, and it is in the form required by the transfer principle, so it is also true for the hyperreal numbers. This shows that it is not possible to use a generic symbol such as ∞ for all the infinite quantities in the hyperreal system; infinite quantities differ in magnitude from other infinite quantities, and infinitesimals from other infinitesimals.

Similarly, the casual use of 1/0 = ∞ is invalid, since the transfer principle applies to the statement that division by zero is undefined. The rigorous counterpart of such a calculation would be that if ε is a non-zero infinitesimal, then 1/ε is infinite.

For any finite hyperreal number x, its standard part, st(x), is defined as the unique real number that differs from it only infinitesimally. The derivative of a function y(x) is defined not as dy/dx but as the standard part of the corresponding difference quotient.

For example, to find the derivative f′(x) of the function f(x) = x², let dx be a non-zero infinitesimal. Then,

${\displaystyle f'(x)}$	${\displaystyle =\operatorname {st} \left({\frac {f(x+dx)-f(x)}{dx}}\right)}$
	${\displaystyle =\operatorname {st} \left({\frac {x^{2}+2x\cdot dx+(dx)^{2}-x^{2}}{dx}}\right)}$
	${\displaystyle =\operatorname {st} \left({\frac {2x\cdot dx+(dx)^{2}}{dx}}\right)}$
	${\displaystyle =\operatorname {st} \left({\frac {2x\cdot dx}{dx}}+{\frac {(dx)^{2}}{dx}}\right)}$
	${\displaystyle =\operatorname {st} \left(2x+dx\right)}$
	${\displaystyle =2x}$

The use of the standard part in the definition of the derivative is a rigorous alternative to the traditional practice of neglecting the square of an infinitesimal quantity. Dual numbers are a number system based on this idea. After the third line of the differentiation above, the typical method from Newton through the 19th century would have been simply to discard the dx² term. In the hyperreal system, dx² ≠ 0, since dx is nonzero, and the transfer principle can be applied to the statement that the square of any nonzero number is nonzero. However, the quantity dx² is infinitesimally small compared to dx; that is, the hyperreal system contains a hierarchy of infinitesimal quantities.

Integration

One way of defining a definite integral in the hyperreal system is as the standard part of an infinite sum on a hyperfinite lattice defined as a, a + dx, a + 2dx, ..., a + ndx, where dx is infinitesimal, n is an infinite hypernatural, and the lower and upper bounds of integration are a and b = a + n dx.

Properties

The hyperreals *R form an ordered field containing the reals R as a subfield. Unlike the reals, the hyperreals do not form a standard metric space, but by virtue of their order they carry an order topology.

The use of the definite article the in the phrase the hyperreal numbers is somewhat misleading in that there is not a unique ordered field that is referred to in most treatments. However, a 2003 paper by Vladimir Kanovei and Saharon Shelah shows that there is a definable, countably saturated (meaning ω-saturated, but not, of course, countable) elementary extension of the reals, which therefore has a good claim to the title of the hyperreal numbers. Furthermore, the field obtained by the ultrapower construction from the space of all real sequences, is unique up to isomorphism if one assumes the continuum hypothesis.

The condition of being a hyperreal field is a stronger one than that of being a real closed field strictly containing R. It is also stronger than that of being a superreal field in the sense of Dales and Woodin.

Development

The hyperreals can be developed either axiomatically or by more constructively oriented methods. The essence of the axiomatic approach is to assert (1) the existence of at least one infinitesimal number, and (2) the validity of the transfer principle. In the following subsection we give a detailed outline of a more constructive approach. This method allows one to construct the hyperreals if given a set-theoretic object called an ultrafilter, but the ultrafilter itself cannot be explicitly constructed.

From Leibniz to Robinson

When Newton and (more explicitly) Leibniz introduced differentials, they used infinitesimals and these were still regarded as useful by later mathematicians such as Euler and Cauchy. Nonetheless these concepts were from the beginning seen as suspect, notably by George Berkeley. Berkeley's criticism centered on a perceived shift in hypothesis in the definition of the derivative in terms of infinitesimals (or fluxions), where dx is assumed to be nonzero at the beginning of the calculation, and to vanish at its conclusion (see Ghosts of departed quantities for details). When in the 1800s calculus was put on a firm footing through the development of the (ε, δ)-definition of limit by Bolzano, Cauchy, Weierstrass, and others, infinitesimals were largely abandoned, though research in non-Archimedean fields continued (Ehrlich 2006).

However, in the 1960s Abraham Robinson showed how infinitely large and infinitesimal numbers can be rigorously defined and used to develop the field of nonstandard analysis. Robinson developed his theory nonconstructively, using model theory; however it is possible to proceed using only algebra and topology, and proving the transfer principle as a consequence of the definitions. In other words hyperreal numbers per se, aside from their use in nonstandard analysis, have no necessary relationship to model theory or first order logic, although they were discovered by the application of model theoretic techniques from logic. Hyper-real fields were in fact originally introduced by Hewitt (1948) by purely algebraic techniques, using an ultrapower construction.

The ultrapower construction

We are going to construct a hyperreal field via sequences of reals. In fact we can add and multiply sequences componentwise; for example:

${\displaystyle (a_{0},a_{1},a_{2},\ldots )+(b_{0},b_{1},b_{2},\ldots )=(a_{0}+b_{0},a_{1}+b_{1},a_{2}+b_{2},\ldots )}$

and analogously for multiplication. This turns the set of such sequences into a commutative ring, which is in fact a real algebra A. We have a natural embedding of R in A by identifying the real number r with the sequence (r, r, r, …) and this identification preserves the corresponding algebraic operations of the reals. The intuitive motivation is, for example, to represent an infinitesimal number using a sequence that approaches zero. The inverse of such a sequence would represent an infinite number. As we will see below, the difficulties arise because of the need to define rules for comparing such sequences in a manner that, although inevitably somewhat arbitrary, must be self-consistent and well defined. For example, we may have two sequences that differ in their first n members, but are equal after that; such sequences should clearly be considered as representing the same hyperreal number. Similarly, most sequences oscillate randomly forever, and we must find some way of taking such a sequence and interpreting it as, say, ${\displaystyle 7+\epsilon }$, where ${\displaystyle \epsilon }$ is a certain infinitesimal number.

Comparing sequences is thus a delicate matter. We could, for example, try to define a relation between sequences in a componentwise fashion:

${\displaystyle (a_{0},a_{1},a_{2},\ldots )\leq (b_{0},b_{1},b_{2},\ldots )\iff (a_{0}\leq b_{0})\wedge (a_{1}\leq b_{1})\wedge (a_{2}\leq b_{2})\ldots }$

but here we run into trouble, since some entries of the first sequence may be bigger than the corresponding entries of the second sequence, and some others may be smaller. It follows that the relation defined in this way is only a partial order. To get around this, we have to specify which positions matter. Since there are infinitely many indices, we don't want finite sets of indices to matter. A consistent choice of index sets that matter is given by any free ultrafilter U on the natural numbers; these can be characterized as ultrafilters that do not contain any finite sets. (The good news is that Zorn's lemma guarantees the existence of many such U; the bad news is that they cannot be explicitly constructed.) We think of U as singling out those sets of indices that "matter": We write (a₀, a₁, a₂, ...) ≤ (b₀, b₁, b₂, ...) if and only if the set of natural numbers { n : a_n ≤ b_n } is in U.

This is a total preorder and it turns into a total order if we agree not to distinguish between two sequences a and b if a ≤ b and b ≤ a. With this identification, the ordered field *R of hyperreals is constructed. From an algebraic point of view, U allows us to define a corresponding maximal ideal I in the commutative ring A (namely, the set of the sequences that vanish in some element of U), and then to define *R as A/I; as the quotient of a commutative ring by a maximal ideal, *R is a field. This is also notated A/U, directly in terms of the free ultrafilter U; the two are equivalent. The maximality of I follows from the possibility of, given a sequence a, constructing a sequence b inverting the non-null elements of a and not altering its null entries. If the set on which a vanishes is not in U, the product ab is identified with the number 1, and any ideal containing 1 must be A. In the resulting field, these a and b are inverses.

The field A/U is an ultrapower of R. Since this field contains R it has cardinality at least that of the continuum. Since A has cardinality

${\displaystyle (2^{\aleph _{0}})^{\aleph _{0}}=2^{\aleph _{0}^{2}}=2^{\aleph _{0}},}$

it is also no larger than ${\displaystyle 2^{\aleph _{0}}}$, and hence has the same cardinality as R.

One question we might ask is whether, if we had chosen a different free ultrafilter V, the quotient field A/U would be isomorphic as an ordered field to A/V. This question turns out to be equivalent to the continuum hypothesis; in ZFC with the continuum hypothesis we can prove this field is unique up to order isomorphism, and in ZFC with the negation of continuum hypothesis we can prove that there are non-order-isomorphic pairs of fields that are both countably indexed ultrapowers of the reals.

For more information about this method of construction, see ultraproduct.

An intuitive approach to the ultrapower construction

The following is an intuitive way of understanding the hyperreal numbers. The approach taken here is very close to the one in the book by Goldblatt. Recall that the sequences converging to zero are sometimes called infinitely small. These are almost the infinitesimals in a sense; the true infinitesimals include certain classes of sequences that contain a sequence converging to zero.

Let us see where these classes come from. Consider first the sequences of real numbers. They form a ring, that is, one can multiply, add and subtract them, but not necessarily divide by a non-zero element. The real numbers are considered as the constant sequences, the sequence is zero if it is identically zero, that is, a_n = 0 for all n.

In our ring of sequences one can get ab = 0 with neither a = 0 nor b = 0. Thus, if for two sequences ${\displaystyle a,b}$ one has ab = 0, at least one of them should be declared zero. Surprisingly enough, there is a consistent way to do it. As a result, the equivalence classes of sequences that differ by some sequence declared zero will form a field, which is called a hyperreal field. It will contain the infinitesimals in addition to the ordinary real numbers, as well as infinitely large numbers (the reciprocals of infinitesimals, including those represented by sequences diverging to infinity). Also every hyperreal that is not infinitely large will be infinitely close to an ordinary real, in other words, it will be the sum of an ordinary real and an infinitesimal.

This construction is parallel to the construction of the reals from the rationals given by Cantor. He started with the ring of the Cauchy sequences of rationals and declared all the sequences that converge to zero to be zero. The result is the reals. To continue the construction of hyperreals, consider the zero sets of our sequences, that is, the ${\displaystyle z(a)=\{i:a_{i}=0\}}$, that is, ${\displaystyle z(a)}$ is the set of indexes ${\displaystyle i}$ for which ${\displaystyle a_{i}=0}$. It is clear that if ${\displaystyle ab=0}$, then the union of ${\displaystyle z(a)}$ and ${\displaystyle z(b)}$ is N (the set of all natural numbers), so:

1. One of the sequences that vanish on two complementary sets should be declared zero
2. If ${\displaystyle a}$ is declared zero, ${\displaystyle ab}$ should be declared zero too, no matter what ${\displaystyle b}$ is.
3. If both ${\displaystyle a}$ and ${\displaystyle b}$ are declared zero, then ${\displaystyle a^{2}+b^{2}}$ should also be declared zero.

Now the idea is to single out a bunch U of subsets X of N and to declare that ${\displaystyle a=0}$ if and only if ${\displaystyle z(a)}$ belongs to U. From the above conditions one can see that:

1. From two complementary sets one belongs to U
2. Any set having a subset that belongs to U, also belongs to U.
3. An intersection of any two sets belonging to U belongs to U.
4. Finally, we do not want the empty set to belong to U because then everything would belong to U, as every set has the empty set as a subset.

Any family of sets that satisfies (2–4) is called a filter (an example: the complements to the finite sets, it is called the Fréchet filter and it is used in the usual limit theory). If (1) also holds, U is called an ultrafilter (because you can add no more sets to it without breaking it). The only explicitly known example of an ultrafilter is the family of sets containing a given element (in our case, say, the number 10). Such ultrafilters are called trivial, and if we use it in our construction, we come back to the ordinary real numbers. Any ultrafilter containing a finite set is trivial. It is known that any filter can be extended to an ultrafilter, but the proof uses the axiom of choice. The existence of a nontrivial ultrafilter (the ultrafilter lemma) can be added as an extra axiom, as it is weaker than the axiom of choice.

Now if we take a nontrivial ultrafilter (which is an extension of the Fréchet filter) and do our construction, we get the hyperreal numbers as a result.

If ${\displaystyle f}$ is a real function of a real variable ${\displaystyle x}$ then ${\displaystyle f}$ naturally extends to a hyperreal function of a hyperreal variable by composition:

${\displaystyle f(\{x_{n}\})=\{f(x_{n})\}}$

where ${\displaystyle \{\dots \}}$ means "the equivalence class of the sequence ${\displaystyle \dots }$ relative to our ultrafilter", two sequences being in the same class if and only if the zero set of their difference belongs to our ultrafilter.

All the arithmetical expressions and formulas make sense for hyperreals and hold true if they are true for the ordinary reals. It turns out that any finite (that is, such that ${\displaystyle |x|<a}$ for some ordinary real ${\displaystyle a}$) hyperreal ${\displaystyle x}$ will be of the form ${\displaystyle y+d}$ where ${\displaystyle y}$ is an ordinary (called standard) real and ${\displaystyle d}$ is an infinitesimal. It can be proven by bisection method used in proving the Bolzano-Weierstrass theorem, the property (1) of ultrafilters turns out to be crucial.

Properties of infinitesimal and infinite numbers

The finite elements F of *R form a local ring, and in fact a valuation ring, with the unique maximal ideal S being the infinitesimals; the quotient F/S is isomorphic to the reals. Hence we have a homomorphic mapping, st(x), from F to R whose kernel consists of the infinitesimals and which sends every element x of F to a unique real number whose difference from x is in S; which is to say, is infinitesimal. Put another way, every finite nonstandard real number is "very close" to a unique real number, in the sense that if x is a finite nonstandard real, then there exists one and only one real number st(x) such that x – st(x) is infinitesimal. This number st(x) is called the standard part of x, conceptually the same as x to the nearest real number. This operation is an order-preserving homomorphism and hence is well-behaved both algebraically and order theoretically. It is order-preserving though not isotonic; i.e. ${\displaystyle x\leq y}$ implies ${\displaystyle \operatorname {st} (x)\leq \operatorname {st} (y)}$, but ${\displaystyle x<y}$ does not imply ${\displaystyle \operatorname {st} (x)<\operatorname {st} (y)}$.

• We have, if both x and y are finite,
  $${\displaystyle \operatorname {st} (x+y)=\operatorname {st} (x)+\operatorname {st} (y)}$$

  
  $${\displaystyle \operatorname {st} (xy)=\operatorname {st} (x)\operatorname {st} (y)}$$

  
• If x is finite and not infinitesimal.
  $${\displaystyle \operatorname {st} (1/x)=1/\operatorname {st} (x)}$$

  
• x is real if and only if
  $${\displaystyle \operatorname {st} (x)=x}$$

  

The map st is continuous with respect to the order topology on the finite hyperreals; in fact it is locally constant.

Hyperreal fields

Suppose X is a Tychonoff space, also called a T_3.5 space, and C(X) is the algebra of continuous real-valued functions on X. Suppose M is a maximal ideal in C(X). Then the factor algebra A = C(X)/M is a totally ordered field F containing the reals. If F strictly contains R then M is called a hyperreal ideal (terminology due to Hewitt (1948)) and F a hyperreal field. Note that no assumption is being made that the cardinality of F is greater than R; it can in fact have the same cardinality.

An important special case is where the topology on X is the discrete topology; in this case X can be identified with a cardinal number κ and C(X) with the real algebra R^κ of functions from κ to R. The hyperreal fields we obtain in this case are called ultrapowers of R and are identical to the ultrapowers constructed via free ultrafilters in model theory.

Ten Lost Tribes

From Wikipedia, the free encyclopedia

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

The ten lost tribes were the ten of the Twelve Tribes of Israel that were said to have been exiled from the Kingdom of Israel after its conquest by the Neo-Assyrian Empire c. 722 BCE. These are the tribes of Reuben, Simeon, Dan, Naphtali, Gad, Asher, Issachar, Zebulun, Manasseh, and Ephraim; all but Judah and Benjamin (as well as some members of Levi, the priestly tribe, which did not have its own territory). The Jewish historian Josephus (37–100 CE) wrote that "there are but two tribes in Asia and Europe subject to the Romans, while the ten tribes are beyond Euphrates till now, and are an immense multitude, and not to be estimated by numbers".

In the 7th and 8th centuries CE, the return of the lost tribes was associated with the concept of the coming of the messiah. Claims of descent from the "lost tribes" have been proposed in relation to many groups, and some religions espouse a messianic view that the tribes will return.

Historians have generally concluded the tribes assimilated into the local population, but this has not stopped various religions from asserting that some survived as distinct entities. Zvi Ben-Dor Benite, a professor of Middle Eastern history, states: "The fascination with the tribes has generated, alongside ostensibly nonfictional scholarly studies, a massive body of fictional literature and folktale." Anthropologist Shalva Weil has documented various differing tribes and peoples claiming affiliation to the Lost Tribes throughout the world.

Scriptural basis

See also: Assyrian captivity and Resettlement policy of the Neo-Assyrian Empire

 

Delegation of the Northern Kingdom of Israel, bearing gifts to the Assyrian ruler Shalmaneser III, c. 840 BCE, on the Black Obelisk, British Museum.

The scriptural basis for the idea of lost tribes is 2 Kings 17:6: "In the ninth year of Hoshea, the king of Assyria took Samaria, and carried Israel away unto Assyria, and placed them in Halah, and in Habor, on the river of Gozan, and in the cities of the Medes."

According to the Bible, the Kingdom of Israel and Kingdom of Judah were the successor states to the older United Monarchy of Israel. The Kingdom of Israel came into existence in about the 930s BCE after the northern tribes of Israel rejected Solomon's son Rehoboam as their king. Nine tribes formed the Kingdom of Israel, the tribes of Reuben, Issachar, Zebulun, Dan, Naphtali, Gad, Asher, Ephraim, and Manasseh.

The tribes of Judah and Benjamin remained loyal to Rehoboam, and formed the Kingdom of Judah. In addition, members of the Tribe of Levi were located in cities in both kingdoms. According to 2 Chronicles 15:9, members of the tribes of Ephraim, Manasseh, and Simeon fled to Judah during the reign of Asa of Judah (c. 911–870 BCE).

In c. 732 BCE, the Assyrian king Tiglath-Pileser III sacked Damascus and Israel, annexing Aramea and territory of the tribes of Reuben, Gad and Manasseh in Gilead including the desert outposts of Jetur, Naphish, and Nodab. People from these tribes were taken captive and resettled in the region of the Khabur River system in Assyria/Mesopotamia. Tiglath-Pilesar also captured the territory of Naphtali and the city of Janoah in Ephraim, and an Assyrian governor was placed over the region of Naphtali. According to 2 Kings 16:9 and 15:29, the population of Aram and the annexed part of Israel was deported to Assyria.

Israel Finkelstein estimated that only a fifth of the population (about 40,000) were actually resettled out of the area during the two deportation periods under Tiglath-Pileser III, Shalmaneser V and Sargon II. Many also fled south to Jerusalem, which appears to have expanded in size fivefold during this period, requiring a new wall to be built, and a new source of water (Siloam) to be provided by King Hezekiah. Furthermore, 2 Chronicles 30:1–11 explicitly mentions northern Israelites who had been spared by the Assyrians—in particular, members of Dan, Ephraim, Manasseh, Asher, and Zebulun—and how members of the latter three returned to worship at the Temple in Jerusalem at that time.

The story of Anna on the occasion of the Presentation of Jesus at the Temple in the New Testament names her as being of the (lost) tribe of Asher (Luke 2:36).

The Hebrew Bible does not use the phrase "ten lost tribes", leading some to question the number of tribes involved. 1 Kings 11:31 states that the kingdom would be taken from Solomon and ten tribes given to Jeroboam:

And he said to Jeroboam, Take thee ten pieces: for thus saith the LORD, the God of Israel, Behold, I will rend the kingdom out of the hand of Solomon, and will give ten tribes to thee.

— 1 Kings 11:31

But I will take the kingdom out of his son's hand, and will give it unto thee, even ten tribes.

— 1 Kings 11:35

Biblical apocrypha

According to Zvi Ben-Dor Benite:

Centuries after their disappearance, the ten lost tribes sent an indirect but vital sign ... In 2 Esdras, we read about the ten tribes and "their long journey through that region, which is called Arzareth" ... The book of the "Vision of Ezra", or Esdras, was written in Hebrew or Aramaic by a Jew in Israel sometime before the end of the first century CE, shortly after the destruction of the temple by the Romans [in 70 CE]. It is one of a group of texts later designated as the so-called Apocrypha—pseudoepigraphal books – attached to but not included in the Hebrew biblical canon.

Views

Judaism

There are discussions in the Talmud as to whether the ten lost tribes will eventually be reunited with the Tribe of Judah; that is, with the Jewish people. In the Talmud, Tractate Sanhedrin equates the exile of the lost tribes with being morally and spiritually lost. In Tractate Sanhedrin 110B, Rabbi Eliezer states:

Just like a day is followed by darkness, and the light later returns, so too, although it will become 'dark' for the ten tribes, God will ultimately take them out of their darkness.

In the Jerusalem Talmud, Rabbi Shimon ben Yehudah, of the town of Acco, states in the name of Rabbi Shimon:

If their deeds are as this day's, they will not return; otherwise they shall.

An Ashkenazi Jewish legend speaks of these tribes as Die Roite Yiddelech, "the little red Jews", who were cut off from the rest of Jewry by the legendary river Sambation, "whose foaming waters raise high up into the sky a wall of fire and smoke that is impossible to pass through."

Christianity

To varying degrees, Apocryphal accounts concerning the Lost Tribes, based on biblical accounts, have been produced by Jews and Christians since at least the 17th century. An increased currency of tales relating to lost tribes that occurred in the 17th century was due to the confluence of several factors. According to Tudor Parfitt:

As Michael Pollack shows, Menasseh's argument was based on "three separate and seemingly unrelated sources: a verse from the book of Isaiah, Matteo Ricci's discovery of an old Jewish community in the heart of China and Antonio Montezinos' reported encounter with members of the Lost Tribes in the wilds of South America".

In 1649, Menasseh ben Israel published his book, The Hope of Israel, in Spanish and Latin in Amsterdam; it included Antonio de Montezinos' account of the Lost Tribes in the New World. An English translation was published in London in 1650. In it, Menasseh argued that the native inhabitants of America which were encountered at the time of the European discovery were actually the descendants of the [lost] Ten Tribes of Israel and for the first time, he tried to gain support for the theory from European thinkers and publishers. Menasseh noted how important Montezinos' account was,

for the Scriptures do not tell what people first inhabited those Countries; neither was there mention of them by any, til Christop. Columbus, Americus, Vespacius [sic], Ferdinandus, Cortez [sic], the Marquesse Del Valle [sic], and Franciscus Pizarrus [sic] went thither ...

He wrote on 23 December 1649: "I think that the Ten Tribes live not only there ... but also in other lands scattered everywhere; these never did come back to the Second Temple and they keep till this day still the Jewish Religion ..."

In 1655, Menasseh ben Israel petitioned Oliver Cromwell to allow the Jews to return to England in furtherance of the Messianic goal. (Since the Edict of Expulsion in 1290, Jews had been prohibited by law from living in England.) With the approach of 1666, considered a significant date, Cromwell was allegedly interested in the return of the Jews to England because of the many theories circulating related to millennial thinking about the end of the world. Many of these ideas were fixed upon the year 1666 and the Fifth Monarchy Men who were looking for the return of Jesus as the Messiah; he was expected to establish a final kingdom to rule the physical world for a thousand years. Messianic believers supported Cromwell's Republic in the expectation that it was a preparation for the fifth monarchy—that is, the monarchy that should succeed the Babylonian, Persian, Greek, and Roman world empires.

Latter-day Saint Movement

Main article: House of Joseph (LDS Church)

The Book of Mormon is based on the premise that two families of Israelites escaped from Israel shortly before the sacking of Jerusalem by Nebuchadnezzar, constructed a ship, sailed across the ocean, and arrived in the New World. They are among the ancestors of Native American tribes and the Polynesians. Adherents believe the two founding tribes were called Nephites and Lamanites, that the Nephites obeyed the Law of Moses, practiced Christianity, and that the Lamanites were rebellious. Eventually the Lamanites wiped out the Nephites around CE 400, and they are among the ancestors of Native Americans.

The Church of Jesus Christ of Latter-day Saints (LDS Church) believes in the literal gathering of Israel, and the Church actively preaches the gathering of people from the twelve tribes. "Today Israelites are found in all countries of the world. Many of these people do not know that they are descended from the ancient house of Israel," the church teaches in its basic Gospel Principles manual. "The Lord promised that His covenant people would someday be gathered .... God gathers His children through missionary work. As people come to a knowledge of Jesus Christ, receiving the ordinances of salvation and keeping the associated covenants, they become 'the children of the covenant' (3 Nephi 20:26)."

The church also teaches that

"The power and authority to direct the work of gathering the house of Israel was given to Joseph Smith by the prophet Moses, who appeared in 1836 in the Kirtland Temple. ... The Israelites are to be gathered spiritually first and then physically. They are gathered spiritually as they join The Church of Jesus Christ of Latter-day Saints and make and keep sacred covenants. ... The physical gathering of Israel means that the covenant people will be 'gathered home to the lands of their inheritance, and shall be established in all their lands of promise' (2 Nephi 9:2). The tribes of Ephraim and Manasseh will be gathered in the Americas. The tribe of Judah will return to the city of Jerusalem and the area surrounding it. The ten lost tribes will receive from the tribe of Ephraim their promised blessings (see D&C 133:26–34). ... The physical gathering of Israel will not be complete until the Second Coming of the Savior and on into the Millennium (see Joseph Smith—Matthew 1:37)."

One of their main Articles of Faith, which was written by Joseph Smith, is as follows: "We believe in the literal gathering of Israel and in the restoration of the Ten Tribes; that Zion (the New Jerusalem) will be built upon the American continent; that Christ will reign personally upon the earth; and, that the earth will be renewed and receive its paradisiacal glory." (LDS Articles of Faith #10)

Regarding the Ezekiel 37 prophecy, the church teaches that the Book of Mormon is the stick of Ephraim (or Joseph) mentioned and that the Bible is the stick of Judah, thus comprising two witnesses for Jesus Christ. The church believes the Book of Mormon to be a collection of records by prophets of the ancient Americas, written on plates of gold and translated by Joseph Smith c. 1830. The church considers the Book of Mormon one of the main tools for the spiritual gathering of Israel.

Historical view

Historians generally concluded that the groups which were referred to as the Lost Tribes merged with the local population. For instance, the New Standard Jewish Encyclopedia states: "In historic fact, some members of the Ten Tribes remained in the land of Israel, where apart from the Samaritans some of their descendants long preserved their identity among the Jewish population, others were assimilated, while others were presumably absorbed by the last Judean exiles who in 597–586 BCE were deported to Assyria ... Unlike the Judeans of the southern Kingdom, who survived a similar fate 135 years later, they soon assimilated ..."

Search

The enduring mysteries surrounding the disappearance of the tribes later became a source of numerous largely mythological narratives in recent centuries, with historian Tudor Parfitt arguing "this myth is a vital feature of colonial discourse throughout the long period of European overseas empires, from the beginning of the fifteenth century, until the later half of the twentieth". Along with Prester John, they formed an imaginary for exploration and contact with indigenous peoples in the Age of Discovery and colonialism.

However, Parfitt's other research indicated some possible ethnic links between several older Jewish Diaspora communities in Asia and Africa and the Middle East, especially those established in pre-colonial times. For example, in his Y-DNA studies of males from the Lemba people, Parfitt found a high proportion of paternal Semitic ancestry, DNA that is common to both Arabs and Jews from the Middle East.

His later genetic studies of the Bene Israel of India, the origins of whom were obscure, also concluded that they were predominantly descended from males from the Middle East, largely consistent with their oral histories of origin. These findings subsequently led other Judaising groups, including the Gogodala tribe of Papua New Guinea, to seek help in determining their own origins.

Ethnology and anthropology

Expanded exploration and study of groups throughout the world through archaeology and the new field of anthropology in the late 19th century led to a revival or a reworking of accounts of the Lost Tribes. For instance, because the Mississippian culture's complex earthwork mounds seemed to be beyond the skills of the Native American cultures which were known to European Americans at the time of their discovery, it was theorized that the ancient civilizations which were involved in the mounds' construction were linked to the Lost Tribes. They tried to fit new information which they acquired as the result of archaeological findings into a biblical construct. However, the earthworks across North America have been conclusively linked to various Native groups, and today, archaeologists consider the theory of non-Native origin pseudo-scientific.

Groups which claim descent from the Lost Tribes

Main article: Groups claiming affiliation with Israelites

Pashtuns of Afghanistan and Pakistan

Main article: Theory of Pashtun descent from Israelites

Among the Pashtuns, there is a tradition of being descended from the exiled lost tribes of Israel. This tradition was referenced in 19th century western scholarship and it was also incorporated in the "Lost Tribes" literature which was popular at that time (notably George Moore's The Lost Tribes of 1861). Recently (2000s), interest in the topic has been revived by the Jerusalem-based anthropologist Shalva Weil, who was quoted in the popular press as stating that the "Taliban may be descended from Jews".

The traditions surrounding the Pashtuns being the remote descendants of the "Lost Tribes of Israel" are to be distinguished from the historical existence of the Jewish community in eastern Afghanistan or northwest Pakistan which flourished from about the 7th century to the early 20th century, but has essentially disappeared from the region due to emigration to Israel since the 1950s.

Mughal-era historiography

Main article: Nimat Allah al-Harawi

According to the Encyclopaedia of Islam, the theory of Pashtun descent from Israelites can be traced to Makhzan-e-Afghani, a history book which was compiled for Khan-e-Jehan Lodhi in the reign of the Mughal Emperor Jehangir in the 17th century.

Modern findings

The Pashtuns are a predominantly Sunni Muslim Iranic people, native to southern Afghanistan and northwestern Pakistan, who adhere to an indigenous and pre-Islamic religious code of honor and culture, Pashtunwali. The belief that the Pashtuns are descended from the lost tribes of Israel has never been substantiated by concrete historical evidence. Many members of the Taliban hail from the Pashtun tribes and they do not necessarily disclaim their alleged Israelite descent.

In Pashto, the tribal name 'Yusef Zai' means the "sons of Joseph".

A number of genetic studies on Jews refute the possibility of a connection, whereas others maintain a link. In 2010, The Guardian reported that the Israeli government was planning to fund a genetic study to test the veracity of a genetic link between the Pashtuns and the lost tribes of Israel. The article stated that "Historical and anecdotal evidence strongly suggests a connection, but definitive scientific proof has never been found. Some leading Israeli anthropologists believe that, of all the many groups in the world which claim to have a connection to the 10 lost tribes, the Pashtuns, or Pathans, have the most compelling case."

Assyrian Jews

Some traditions of the Assyrian Jews claim that Israelites of the tribe of Benjamin first arrived in the area of modern Kurdistan after the Neo-Assyrian Empire's conquest of the Kingdom of Israel during the 8th century BCE; they were subsequently relocated to the Assyrian capital. During the first century BCE, the Assyrian royal house of Adiabene—which, according to the Jewish historian Flavius Josephus, was ethnically Assyrian and whose capital was Erbil (Aramaic: Arbala; Kurdish: Hewlêr)—was converted to Judaism. King Monobazes, his queen Helena, and his son and successor Izates are recorded as the first proselytes.

Kashmiri Jews

Main article: Theory of Kashmiri descent from lost tribes of Israel

According to Al-Biruni, the famous 11th-century Persian Muslim scholar: "In former times the inhabitants of Kashmir used to allow one or two foreigners to enter their country, particularly Jews, but at present they do not allow any Hindus whom they do not know personally to enter, much less other people."

François Bernier, a 17th-century French physician and Sir Francis Younghusband, who explored this region in the 1800s, commented on the similar physiognomy between Kashmiris and Jews, including "fair skin, prominent noses," and similar head shapes.

Baikunth Nath Sharga argues that, despite the etymological similarities between Kashmiri and Jewish surnames, the Kashmiri Pandits are of Indo-Aryan descent while the Jews are of Semitic descent.

Bnei Menashe

Main article: Bnei Menashe

Since the late 20th century, some tribes in the Indian North-Eastern states of Mizoram and Manipur have been claiming that they are Lost Israelites and they have also been studying Hebrew and Judaism. In 2005, the chief rabbi of Israel ruled that the Bnei Menashe are descended from a lost tribe. Based on the ruling, Bnei Menashe are allowed to immigrate to Israel after they formally convert to Judaism. In 2021 4,500 Bnei Menashe had made aliyah to Israel; 6,000 Bnei Menashe in India hope to make aliyah.

Beta Israel of Ethiopia

Main article: Beta Israel

The Beta Israel ("House of Israel") are Ethiopian Jews, who were also called "Falashas" in the past. Some members of the Beta Israel, as well as several Jewish scholars, believe that they are descended from the lost Tribe of Dan, as opposed to the traditional account of their origins which claims that they are descended from the Queen of Sheba and the Israelite king Solomon. They have a tradition of being connected to Jerusalem. Early DNA studies showed that they were descended from Ethiopians, but in the 21st century, new studies have shown their possible descent from a few Jews who lived in either the 4th or 5th century CE, possibly in Sudan. The Beta Israel made contact with other Jewish communities in the later 20th century. In 1973 Rabbi Ovadia Yosef, then the Chief Sephardic Rabbi, based on the Radbaz and other accounts, ruled that the Beta Israel were Jews and should be brought to Israel; two years later that opinion was confirmed by a number of other authorities who made similar rulings, including the Chief Ashkenazi Rabbi Shlomo Goren.

Igbo Jews

Main article: Igbo Jews

The Igbo Jews of Nigeria variously claim descent from the tribes of Ephraim, Naphtali, Menasseh, Levi, Zebulun and Gad. The theory, however, does not hold up to historical scrutiny. Historians have examined the historical literature on West Africa from the colonial era and they have elucidated that such theories served diverse functions for the writers who proposed them.

Speculation regarding other ethnic groups

There has been speculation regarding various ethnic groups, which would be regarded as fringe theories.

Japanese people

Main article: Japanese-Jewish common ancestry theory

Some writers have speculated that the Japanese people may be the direct descendants of some of the Ten Lost Tribes. Parfitt writes that "the spread of the fantasy of Israelite origin ... forms a consistent feature of the Western colonial enterprise. ... It is in fact in Japan that we can trace the most remarkable evolution in the Pacific of an imagined Judaic past. As elsewhere in the world, the theory that aspects of the country were to be explained via an Israelite model was introduced by Western agents."

In 1878, Scottish immigrant to Japan Nicholas McLeod self-published Epitome of the Ancient History of Japan. McLeod drew correlations between his observations of Japan and the fulfillment of biblical prophecy: The civilized race of the Aa. Inus,^{[sic: read Ainus]} the Tokugawa and the Machi No Hito of the large towns, by dwelling in the tent or tabernacle shaped houses first erected by Jin Mu Tenno, have fulfilled Noah's prophecy regarding Japhet, "He shall dwell in the tents of Shem."

Jon Entine emphasizes the fact that DNA evidence shows that there are no genetic links between Japanese and Israelite people.

Lemba people

Main article: Lemba people

The Lemba people (Vhalemba) of Southern Africa claim to be the descendants of several Jewish men who traveled from what is now Yemen to Africa in search of gold, where they took wives and established new communities. They specifically adhere to religious practices which are similar to those which exist in Judaism and they also have a tradition of being a migrant people, with clues which point to their origin in either West Asia or North Africa. According to the oral history of the Lemba, their ancestors were Jews who came from a place called Sena several hundred years ago and settled in East Africa. Sena is an abandoned ancient town in Yemen, located in the eastern Hadramaut valley, which history indicates was inhabited by Jews in past centuries. Some research suggests that "Sena" may refer to Wadi Masilah (near Sayhut) in Yemen, often called Sena, or alternatively to the city of Sana'a, which is also located in Yemen.

Māori

Some early Christian missionaries to New Zealand speculated that the native Maori were descendants of the Lost Tribes. Some Māori later embraced this belief.

Native Americans

See also: Northern Cherokee Nation of the Old Louisiana Territory § Middle Eastern origin stories

In 1650, a British minister named Thomas Thorowgood, who was a preacher in Norfolk, published a book entitled Jewes in America or Probabilities that the Americans are of that Race, which he had prepared for the New England missionary society. Parfitt writes of this work: "The society was active in trying to convert the Indians but suspected that they might be Jews and realized that it had better be prepared for an arduous task. Thorowgood's tract argued that the native populations of North America were descendants of the Ten Lost Tribes."

In 1652 Hamon L'Estrange, an English author who wrote literary works about topics such as history and theology published an exegetical tract called Americans no Jews, or improbabilities that the Americans are of that Race in response to the tract by Thorowgood. In response to L'Estrange, in 1660, Thorowgood published a second edition of his book with a revised title and a foreword which was written by John Eliot, a Puritan missionary to the Indians who had translated the Bible into an Indian language.

The American diplomat and journalist Mordecai Manuel Noah also proposed the idea that the indigenous peoples of the Americas are descended from the Israelites in his publication The American Indians Being the Descendants of the Lost Tribes of Israel (1837).

That some or all American Indians are part of the lost tribes is suggested by the Book of Mormon (1830) and it is also a popular belief among Latter-day Saints.

Scythian/Cimmerian theories and British Israelism

Main article: British Israelism

 

A depiction of either King Jehu, or Jehu's ambassador, kneeling at the feet of Shalmaneser III on the Black Obelisk.

Adherents of British Israelism and Christian Identity both believe that the lost tribes migrated northward, over the Caucasus, and became the Scythians, Cimmerians and Goths, as well as the progenitors of the later Germanic invaders of Britain.

The theory first arose in England and then it spread to the United States. During the 20th century, British Israelism was promoted by Herbert W. Armstrong, founder of the Worldwide Church of God.

Tudor Parfitt, author of The Lost Tribes: The History of a Myth, states that the proof which is cited by adherents of British Israelism is "of a feeble composition even by the low standards of the genre," and these notions are widely rejected by historians.

In literature

In 1929 Lazar Borodulin published the only  Yiddish science fiction novel, Yiddish: אויף יענער זייט סמבטיון : וויסענשאפטליכער און פאנטאסטישער ראמאן, romanized: Oyf yener zayt sambatyen, visnshaftlekher un fantastisher roman (On the other side of the Sambation, a scientific and fantastic novel), a novel in the "lost world" genre, written in a Jewish perspective. In the novel a journalist meets a mad scientist with a ray gun in the land of the Red Jews.

In a 1934 Ben Aronin's adventure novel The Lost Tribe. Being the Strange Adventures of Raphael Drale in Search of the Lost Tribes of Israel, a teenager, Raphael, finds the lost tribe of Dan beyond the Arctic Circle.