Re: True = [ proven | provable ]

From: Torkel Franzen (torkel_at_sm.luth.se)
Date: 01/24/05 

Date: 24 Jan 2005 08:59:03 +0100

D.McAnally@i'm_a_gnu.uq.net.au (David McAnally) writes:

> I had thought that the formula of a language (and, in particular, the
> sentences) could be well-ordered, once you have a well-ordering of the
> constant symbols, function symbols, relation symbols, and variables. I
> had also thought that a well-ordering of the sentences would be sufficient
> to prove that every consistent set of sentences has a consistent complete
> extension.
>
> As a consequence, I had thought that the axiom of choice would only be
> needed to guarantee a well-ordering of constant symbols, function symbols
> and relation symbols.

  Right, that's one way of doing it. Whichever way we go about it, the
axiom of choice enters into showing that every consistent set of
sentences has a consistent complete extension.

> In other words, I had thought that, in the general
> case, the axiom of choice was needed to introduce the witnesses (the axiom
> of choice being specifically needed to well-order the symbols of the
> (extended) language).

  We don't need choice to introduce the witnesses, just a sufficient
supply of new constants. Given our language L, first introduce a new
constant c_A for every formula A in L (for this, we don't need
choice). Repeat the procedure omega times, take the union L' of the
resulting languages and add axioms (Ex)A(x)->A(c) where c is the
constant associated with A(x).

Relevant Pages

  • Re: True = [ proven | provable ]
    ... > constant symbols, function symbols, relation symbols, and variables. ... > had also thought that a well-ordering of the sentences would be sufficient ... > to prove that every consistent set of sentences has a consistent complete ... I had thought that the axiom of choice would only be ...
    (sci.math)
  • Re: True = [ proven | provable ]
    ... > constant symbols, function symbols, relation symbols, and variables. ... > had also thought that a well-ordering of the sentences would be sufficient ... > to prove that every consistent set of sentences has a consistent complete ... I had thought that the axiom of choice would only be ...
    (sci.logic)
  • Re: True = [ proven | provable ]
    ... >sentences has a consistent complete extension. ... had also thought that a well-ordering of the sentences would be sufficient ... to prove that every consistent set of sentences has a consistent complete ... I had thought that the axiom of choice would only be ...
    (sci.math)
  • Re: True = [ proven | provable ]
    ... >sentences has a consistent complete extension. ... had also thought that a well-ordering of the sentences would be sufficient ... to prove that every consistent set of sentences has a consistent complete ... I had thought that the axiom of choice would only be ...
    (sci.logic)
  • Re: True = [ proven | provable ]
    ... >sentences has a consistent complete extension. ... had also thought that a well-ordering of the sentences would be sufficient ... to prove that every consistent set of sentences has a consistent complete ... I had thought that the axiom of choice would only be ...
    (comp.theory)
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