Re: what makes it true?


>> I can understand, given a formal system, whether I have a well formed
>> formula, a proof, etc because I can apply whatever rules are given
>> for forming formulae or for deriving new strings from the ones already
>> obtained. However, without a set theory, I can't talk about the set of
>> statements or prove anything about proofs. So my understanding of
>> formal systems would be purely local if not for some type of set theory.

> All of this sounds very strange. I don't see how it relates to
>anything that actually happens. To begin with, what do you mean by
>being given a formal system? In the real world, this means being
>given such explanations as "A->(B->A) is an axiom for all formulas
>A and B". We understand such explanations perfectly well without
>any set theory whatsoever, and I can't imagine by which procedure
>one would first introduce a set theory and then somehow use that
>set theory to elucidate the explanation.

Like I said, I can follow the rules of the system without having a set
theory. However, to prove anything about the system, such as
consistency, I have to be able to talk about the set of statements
in the theory, so I need a set theory. To be able to talk about the set
of provable statements, I'll need to actually talk about sets. To talk
about truth in the theory, I need to talk about models of the theory.
Since a model is a set, that requires a set theory. To ask whether
a statement is independent from the theory, I need to be able to talk
about whether it is in the set of provable statements from the theory.
I don't see any way around it.

--Dan Grubb
.

Relevant Pages

  • Re: what makes it true?
    ... grubb@xxxxxxxxxxxxxxxxx (Daniel Grubb) writes: ... > I can understand, given a formal system, whether I have a well formed ... > formal systems would be purely local if not for some type of set theory. ... All of this sounds very strange. ...
    (sci.math)
  • Re: Request for Reference/Link to example of defining a theory/logic.
    ... that will emulate first-order logic and set theory. ... Long story not worth going into here. ... is incomplete and therefore I need a new improved formal system. ...
    (sci.logic)
  • Re: Request for Reference/Link to example of defining a theory/logic.
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    (sci.logic)
  • Re: Epistemology 201: The Science of Science
    ... I would say that mathematics includes set theory, ... >>are not at odds with its formal system. ... So note that the rules of inference tend to describe the /form/ ...
    (sci.math)
  • Re: Epistemology 201: The Science of Science
    ... I would say that mathematics includes set theory, ... >>are not at odds with its formal system. ... So note that the rules of inference tend to describe the /form/ ...
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