[FOM] Infinity and the "Noble Lie"
Jan Mycielski
jmyciel at euclid.colorado.edu
Sat Jan 7 19:19:56 EST 2006
Mycielski:
Dear Haney and Shipman,
I feel that Your positions are too radical, or insuficiently descriptive of
reality.
The axiom of infinity is not so distant from reality as to claim that it is
nonsense (Haney), and it is not so unambiguous as to claim that it is true
(Shipman).
I believe that a correct (brief) description of the situation is the
following:
Nature (evolution) endowed us with the rules and axioms of logic and set
theory as a framework to analyse and describe reality (to make useful
predictions). That is visible from the fact that all mathematicians know how
to use them correctly in their definitions and proofs, and the few among
them who learned these axioms and rules feel in a strong sense that they are
obvious (with the exception of a tiny minority that was raised in cages and
failed to develop their wings). That natural framework contains the axiom of
infinity, in as much as it is natural to think in terms of unending discrete
processes and continua, and it is natural to think that space-time is
continuous (and hence infinite).
On the other hand we cannot talk honestly about the truth (in the usual
sense of the word true) of any statement unless this statement refers in a
clear way to a real object or process (real as opposed to imaginary). And of
course the axiom of infinity does not refer to any such thing. [However, we
do have a mathematical theory of that natutal relation of truth (due to
Tarski), with his famous undefinability theorem.]
Jan Mycielski
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