Search All of the Math Forum:
Views expressed in these public forums are not endorsed by
NCTM or The Math Forum.


Math Forum
»
Discussions
»
sci.math.*
»
sci.math
Notice: We are no longer accepting new posts, but the forums will continue to be readable.
Topic:
Formally Unknowability, or absolute Undecidability, of certain arithmetic formulas.
Replies:
22
Last Post:
Jan 29, 2013 8:21 PM



fom
Posts:
1,968
Registered:
12/4/12


Re: Formally Unknowability, or absolute Undecidability, of certainarithmeticformulas.
Posted:
Jan 29, 2013 4:41 AM


On 1/28/2013 11:28 PM, Nam Nguyen wrote: > On 28/01/2013 6:20 AM, Frederick Williams wrote: >> Nam Nguyen wrote: >> >>> I meant, what would "tomorrow", "today" have anything to to with >>> _mathematical logic_ ? >> >> Oh, a lot. Look up 'temporal logic'. In my day it was something of a >> curiosity of interest only to philosophers (hiss, boo, etc) but now it >> is of much interest to computer scientists among others. > > It seems you aren't aware, but the assumed logic of this thread here > is the familiar FOL=. >
How can that be if you are requesting alternative interpretations of quantification?
However, the answer to your question concerning "tomorrow" and "today" is found in the relationship of model theory to description theory.
Originally, Frege spoke of incomplete symbols such as
x+2=5
because they require a "name" to have a "truth value".
Modern model theory is a bit senseless because they use a parameterized theory (set theory) to justify speaking of "truth" for an object language. If you actually read Tarski's paper, it explicitly excludes consideration of how the "objects" of an interpretation transform incomplete symbols to complete symbols (those with a truth value). This reflects the Russellian position that "naming" is an extralogical function.
One gets to an explicit discussion of names and indentity within a model in Abraham Robinson's "On Constrained Denotation". Whether or not one agrees with Robinson, it returns the question of truth valuation to the role of descriptions and reference.
Having gone this far, the next issue is the relation between demonstratives and descriptions. This involves indexicals. Kaplan produced a decent intensional logic of demonstratives that makes plain the relation between demonstratives and descriptions. Since it utilizes indexicals, temporal modal operators play a role.
To say that
x+2=5
is true because
there exists an "object" y such that
y+2=5
is different from saying that
3+2=5
is true.
That is the difference between using a "set" and a "name".
The history of description theory explains why this is not taught in mathematical logic. But that historical basis has been collapsing for over 50 years. This change has simply been ignored by the mathematical community.



