[FOM] Infinity and the "Noble Lie"

[joeshipman@aol.com]

Shipman:

> For those following this discussion who have no problem with the Axiom
> of Infinity in ZFC, and who are comfortable saying that it is a true
> statement and that statements derived from it are therefore also true,
> I'd like to hear what your attitude is to the axiom that an
> Inaccessible Cardinal exists.

Koskensilta:

It's also true, as are various other small large cardinal axioms up to
something like a weakly compact cardinal at least. What's the point of
this question?

My reply to Koskensilta:

The point of the question is that I don't expect everyone to have the 
same answer as you. What is it about the large cardinals up to a weakly 
compact cardinal that makes you believe they are true?

Tait:

One property of lies, noble or otherwise, is that they are
*falsehoods*; and this presupposes an antecedent notion of truth. One
can question whether there is an antecedent  notion of truth in
mathematics,  which serves as a common ground upon which to resolve
the debate about whether there are infinite sets. It seems to me to
be very analogous to the debate between constructivists and non-
constructivists, where, also, one has to ask: On what none question-
begging grounds could one possibly resolve the issue.

My reply to Tait:

My point is that there does appear to be an antecedent notion of truth 
in mathematics as far as "ordinary mathematics" is concerned --  for 
example, there is a fair consensus that the Riemann Hypothesis is 
either true or false but we don't know which. Those who deny that an 
antecedent notion of truth exists which settles the Axiom of Infinity 
had better explain what to make of statements which do not mention 
infinite sets but all known proofs of which require the axiom of 
infinity. (For example, Kruskal's theorem in Friedman's finite form, or 
various other specializations of the Graph Minor theorem.)

Mani:

My point was that axioms may or may not be called 'admissible'. I did 
not say
anything about the truth. Mathematically, I think the axiom of infinity 
is
perfectly true. The concept of truth of axioms must be seen in a 
hierarchy
defined by the abstraction levels associated. Interpreting mathematics 
in a
particular 'semantic domain' involves many assumptions on the mechanism 
of
interpretation.

Actually mathematics starts from reality and from the truth 
corresspondences
therein we eventually abstract these axioms. So axioms must be true by
definition. Particular abstractions may be favoured over others. It is 
also
not necessary to find a fault with a version to investigate another.
'Absolute truth of ZFC axioms' is all right for mathematicians not 
working on
the foundations.


My reply to Mani:

Your last sentence is exactly how a "noble lie" works -- the statements 
are true as far as the masses are concerned, but we enlightened ones 
know the situation is more complicated....

You're still not addressing my main point. I am not insisting you 
declare that the ZFC axioms are "true" in a context-independent sense; 
I am asking whether ANY theorem whose known ZFC-proofs require the 
Axiom of Infinity can be "true" in a context-independent sense.