FOM: Goedel: truth and misinterpretations

Torkel Franzen torkel at
Wed Oct 25 02:32:42 EDT 2000

  V.Kanovei says:

  >Let's forget Goedel's mystical "sentences" and consider CH 
  >which everybody knows what it means (at least formally). 
  >neither CH nor not-CH is a theorem of ZFC -- and this is 
  >a mathematical fact (I let aside the assumption of Con ZFC).

  What's mystical about arithmetical sentences? Do you have any objection
to the assertion that (assuming ZFC to be consistent) there are infinitely
many true sentences of the form "the Diophantine equation P=0 has no
solution" which are not provable in ZFC?

  >Now, the misinterpretation which I am talking about reduces 
  >to the following: 
  >BUT either CH or not-CH is TRUE ,
  >which is here nothing but an example of the excludedmiddle. 
  >Now let me ask again: True -- where ? in which sense ?

  Naturally in the ordinary mathematical sense: "CH is true" is a
mathematical statement which is mathematically equivalent to CH itself.
When people emphatically assert "either CH or not-CH is true", they
are using a tautology to make a metaphysical claim (just as when
people say "what will be will be"). To understand this claim we need
to consider what role it plays in their thinking. The meaning of "true"
is not in itself problematic, or any more problematic than mathematical
concepts in general.

  So do you have any objection to the reflection that CH may well be
true although not provable in ZFC, given that "CH is true" is here a
mathematical statement mathematically equivalent to CH?

  >My view is that ZFC concentrates, in the form of a few 
  >simply formulated principles, everything which has been 
  >practiced in mathematics, although, perhaps, in more 
  >generality that 99.9/100 of mathematics really needs.

  Well, whether you use the axioms of ZFC or some other axioms, how would
you justify those axioms?  Not by proofs, surely?

Torkel Franzen, Luleå university

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