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In what connection did Zermelo say: "Could not just this seemingly so fruitful hypothesis of the infinite have introduced straight contradictions into mathematics, thereby destroying the basic nature of this science that is so proud upon its consistency?" (H.-D. Ebbinghaus, V. Peckhaus: "Ernst Zermelo, an approach to his life and work", Springer (2007) p. 292) And what can he have meant by this sentence?

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The quote is given in section 4.4.5 of Ebbinghaus, Peckhaus titled "The Justification Problem", where The Justification Problem is defined as:

the necessity of justifying the existence of (infinite) models

According to Ebbinghaus, Zermelo expresses the view that:

in general it is impossible to prove consistency, i.e., to exhibit a model.

(Note that Zermelo is speaking here in 1929, two years before Gödel proved his incompleteness results, with its implications for the provability of consistency.)

So the "fruitful hypothesis" is the assumption of a consistent model of axiomatic set theory, and the contradictions are those which may be present in any axiomatization in the absence of a consistency proof.


TL/DR;

Zermelo's reasoning is that in a mathematical proof of consistency, we could never be sure that we have taken into account all modes of inference used in mathematics:

Such a demonstration, if it were possible, would have to rest on a thorough and complete formalisation of all the logic relevant to mathematics. Any "incompleteness" in the underlying "proof theory" such as a neglected possible inference would jeopardise the entire proof. But, since such "completeness" could obviously never be guaranteed, it is, in my opinion, not possible to furnish a formal proof of the consistency.

Zermelo goes on to note that a consistency proof would have to deal with an infinite set of sentences and therefore would simply shift the problem from one infinite domain to another.

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The answer to the second question is quite clear: he means that mathematics dealing with sets of infinite cardinality might be inconsistent (i.e. lead to contradictions). As a specific example , we do not know if ZF (the Zermelo-Frankel axiomatic set theory) is consistent, and, more worrisome, the Peano arithmetic might be inconsistent. We still do not know. There are some recent attempts by Ed Nelson to prove inconsistency of Peano's Axioms, but, as far as I know, they were unsuccessful (per wikipedia: "An error was found in the proof, and he retracted the claim"). See also this and this wikipedia articles on finitism and ultrafinitism, which are attempts to avoid dealing with infinities in math.

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  • $\begingroup$ Such a claim needs evidence. Not merely "quite clear". $\endgroup$ Aug 8, 2017 at 0:35
  • $\begingroup$ @GeraldEdgar I guess we just have different notions of clarity then. $\endgroup$ Aug 8, 2017 at 0:57

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