Controversy over Cantor's theory

In mathematical logic, the theory of infinite sets was first developed by Georg Cantor. Although this work has become a thoroughly standard fixture of classical set theory, it has been criticized in several areas by mathematicians and philosophers.

Cantor's theorem is that there are sets having cardinality greater than the (already infinite) cardinality of the set of whole numbers {1, 2, 3, …}.

Cantor's argument

Cantor's 1891 argument is that there exists an infinite set (which he identifies with the set of real numbers) which has a larger number of elements, or, as he put it, has a greater 'mightiness' (Mächtigkeit), than the infinite set of finite whole numbers {1, 2, 3, …}.

There are a number of steps in his argument, as follows:

Cantor presented a well-ordered sequence of cardinal numbers, the alephs, and attempted to prove that the power of every well-defined set ("consistent multiplicity") is an aleph; and therefore that the ordering relation among alephs determines an order among the sizes of sets.[1] However this proof was flawed, and as Zermelo wrote, "It is precisely at this point that the weakness of the proof sketched here lies… It is precisely doubts of this kind that impelled ... [my own] proof of the well-ordering theorem purely upon the axiom of choice…"[1]

The assumption of the axiom of choice was later shown unnecessary by the Schröder–Bernstein theorem, which makes use of the notion of injective functions from one set to another—a correlation which associates different elements of the former set with different elements of the latter set. The theorem shows that if there is an injective function from set A to set B, and another one from B to A, then there is a bijective function from A to B, and so the sets are equipollent, by the definition we have adopted. Thus it makes sense to say that the power of one set is at least as large as another if there is an injection from the latter to the former, and this will be consistent with our definition of having the same power. Since the set of natural numbers can be embedded in its power set, but the two sets are not of the same power, as shown, we can therefore say the set of natural numbers is of lesser power than its power set. However, despite its avoidance of the axiom of choice, the proof of the Cantor-Bernstein-Schröder theorem is still not constructive, in that it does not produce a concrete bijection in general.

Reception of the argument

At the start, Cantor's theory was controversial among mathematicians and (later) philosophers. As Leopold Kronecker claimed: "I don't know what predominates in Cantor's theory – philosophy or theology, but I am sure that there is no mathematics there". Many mathematicians agreed with Kronecker that the completed infinite may be part of philosophy or theology, but that it has no proper place in mathematics. Logician Wilfrid Hodges (1998) has commented on the energy devoted to refuting this "harmless little argument" (i.e. Cantor's diagonal argument) asking, "what had it done to anyone to make them angry with it?"[2] Contrary to Hodges assertion, others have also taken issue with Cantor's proof regarding the cardinality of the power set.[3][4] Mathematician Solomon Feferman has referred to Cantor's theories as “simply not relevant to everyday mathematics.”[5]

Before Cantor, the notion of infinity was often taken as a useful abstraction which helped mathematicians reason about the finite world, for example the use of infinite limit cases in calculus. The infinite was deemed to have at most a potential existence, rather than an actual existence.[6] "Actual infinity does not exist. What we call infinite is only the endless possibility of creating new objects no matter how many exist already".[7] Carl Friedrich Gauss's views on the subject can be paraphrased as: 'Infinity is nothing more than a figure of speech which helps us talk about limits. The notion of a completed infinity doesn't belong in mathematics'.[8] In other words, the only access we have to the infinite is through the notion of limits, and hence, we must not treat infinite sets as if they have an existence exactly comparable to the existence of finite sets.

Cantor's ideas ultimately were largely accepted, strongly supported by David Hilbert, amongst others. Hilbert predicted: "No one will drive us from the paradise which Cantor created for us".[9] To which Wittgenstein replied "if one person can see it as a paradise of mathematicians, why should not another see it as a joke?".[10] The rejection of Cantor's infinitary ideas influenced the development of schools of mathematics such as constructivism and intuitionism.

Objection to the axiom of infinity

Further information: Finitism

A common objection to Cantor's theory of infinite number involves the axiom of infinity (which is, indeed, an axiom and not a logical truth). Mayberry has noted that "The set-theoretical axioms that sustain modern mathematics are self-evident in differing degrees.[11] One of them – indeed, the most important of them, namely Cantor's axiom, the so-called axiom of infinity – has scarcely any claim to self-evidence at all".

Another objection is that the use of infinite sets is not adequately justified by analogy to finite sets. Hermann Weyl wrote:

… classical logic was abstracted from the mathematics of finite sets and their subsets …. Forgetful of this limited origin, one afterwards mistook that logic for something above and prior to all mathematics, and finally applied it, without justification, to the mathematics of infinite sets. This is the Fall and original sin of [Cantor's] set theory …."[12]

The difficulty with finitism is to develop foundations of mathematics using finitist assumptions, that incorporates what everyone would reasonably regard as mathematics (for example, that includes real analysis).

See also

Notes

  1. 1 2 Cantor, letter to Richard Dedekind, with comments by Ernst Zermelo, translated in van Heijenoort, J., From Frege to Gödel, A Source Book in Mathematical Logic, 1879-1931, Harvard University Press, Cambridge, MA, 1967. Reprinted with corrections, 1977.
  2. ↑ Hodges, Wilfrid (1998), "An Editor Recalls Some Hopeless Papers", The Bulletin of Symbolic Logic (Association for Symbolic Logic) 4 (1), pp. 1–16, doi:10.2307/421003, JSTOR 421003
  3. ↑ Perez, Juan A. (2010). "Addressing mathematical inconsistency: Cantor and Gödel refuted". arXiv:1002.4433 [math.GM].
  4. ↑ Zenkin, Alexander. "Cantor's Diagonal Argument: A New Aspect" (PDF). Dorodnitsyn Computing Center. Retrieved 2 October 2014.
  5. ↑ Wolchover, Natalie. "Dispute over Infinity Divides Mathematicians". Scientific American. Retrieved 2 October 2014.
  6. ↑ Zenkin, Alexander (2004), "Logic Of Actual Infinity And G. Cantor's Diagonal Proof Of The Uncountability Of The Continuum", The Review of Modern Logic 9 (30), pp. 27–80
  7. ↑ (Poincaré quoted from Kline 1982)
  8. ↑ Dunham, William. Journey through Genius: The Great Theorems of Mathematics. Penguin. p. 254.
  9. ↑ (Hilbert, 1926)
  10. ↑ (RFM V. 7)
  11. ↑ Mayberry 2000, p. 10
  12. ↑ Weyl, 1946

References

"Aus dem Paradies, das Cantor uns geschaffen, soll uns niemand vertreiben können."
Translated in Van Heijenoort, Jean, On the infinite, Harvard University Press 

External links

This article is issued from Wikipedia - version of the Saturday, March 19, 2016. The text is available under the Creative Commons Attribution/Share Alike but additional terms may apply for the media files.