Re: Goedel - interesting problem?

From: Jeffrey Ketland (ketland_at_ketland.fsnet.co.uk)
Date: 06/19/04 

Date: Sat, 19 Jun 2004 04:23:58 +0100

Spike wrote in message <40d2c63b$0$92045$45beb828@newscene.com>...
>
>"Jeffrey Ketland" <ketland@ketland.fsnet.co.uk> wrote in message
[snip]
> I assume that you posted the above only as a helpful answer to
> Larry's question, so that I must exclude you from my remarks.
> Larry's question, and thus the above definition, fits into the
> larger debate over Dolan's piece by becoming an expansion of the
> word "true" in the phrase:
>
> "...statements which are true in that set of axioms..."

Saying that a statement is "true in that set of axioms" is merely evidence
of confusion and/or ignorance. A statement is not "true in a set of axioms",
because truth has nothing to do with *axioms*. For example, the set Tr of
arithmetic truths, which I gave a definition of in the previous post, has
nothing to do with axioms. In fact, the set Tr of arithmetic truths is not
axiomatizable. This follows from Tarski's Indefinability Theorem.

In general, whether a statement is true depends upon whether the relevant
state of affairs that it expresses holds or not.

      A statement is true iff it says that such-and-such is the
                                  case, and such-and-such is the case.

As Aristotle put it, "To say of what is, that it is, is true". Of course,
whether a linguistic item like a statement is true depends upon its
interpretation. But it should be stressed that in usual cases, this is left
implicit. If someone asks "Was GW Bush's opening statement of his speech
yesterday true?", nobody wonders whether he is using the word "Iraq" to
refer to the planet Venus. He is speaking English, an interpreted language,
in which "Iraq" is a singular term which refers to Iraq.

The language of arithmetic, for example, is an interpreted language---by
definition. So, "0" refers to the number 0, and "+" denotes the addition
operation and the quantifiers range over N, and so on. So, "2+2 = 4" is true
if and only if 2+2=4. One could take the constant "0" to refer to Paul
McCartney if one liked, but this would be a very weird re-interpretation of
the language of arithmetic. Admissible, but weird and pointless.

The statement "John Lennon was born in Liverpool" is true (in its standard
English interpretation), because John Lennon was, in fact, born in
Liverpool. This has nothing to do with "axioms". Similarly, the arithmetic
statement "There are infinitely many primes" is true (in the standard
arithmetic interpretation), because there are, in fact, infinitely many
primes.

These ideas were made precise by the Polish logician Alfred Tarski in a book
published in 1933.

Simplifying somewhat, if we are discussing (in English) the notion of truth
for an *interpreted* object language L, then we always assert instances of
the Tarski T-scheme,

    (T) S is true in L if and only if p

where S is a sentence of L and "p" is replaced by the translation of S into
English.
For example,

    "Schnee ist weiss" is true in German if and only if snow is white,

If the meta-language contains the object language, then the need for a
translation lapses, and we have the "disquotational T-scheme"

      "p" is true if and only if p

To sum up: People who write things like "true in a set of axioms" are merely
displaying their confusion, like someone who doesn't know what a
differentiable function is.

--- Jeff

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