Re: Name the thesis: "Formal sentences capture informal ones"
From: Torkel Franzen (torkel_at_sm.luth.se)
Date: 01/29/05
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Date: 29 Jan 2005 21:38:02 +0100
tchow@lsa.umich.edu writes:
> Then how about, "All informal statements of ordinary mathematics are
> expressible by formal sentences in ZFC"?
The Church-Turing thesis only requires that for every function
computable by an algorithm, there is a Turing machine that computes
the same function. It is not required that the Turing machine embodies
the same algorithm. The above is a lot more problematic to make
precise. What is it we require of a formal sentence in ZFC if it is to
express a given informal statement of ordinary mathematics? In
practice, we recognize formalizations as adequate, but there are many
different ways of formalizing informal mathematical statements, and it
is far from clear how to characterize what counts as an adequate
formalization.
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