[FOM] R: Improving Set Theory

[Antonino Drago]

I try to answer to Friedman answer according to my opinion.

 ST is specific for only one kind of mathematics, that based on the actual
infinity, pre-supposed by the notion of set (or sack in Cantor's analogy)
and the classical logic (as strongly claimed by Hilbert) or equivalently the
organization of the theory as an axiomatic (as ST was since the time of
Zermelo). These are the choices allowing the highest abstractions, which for
a century led mathematicians to hope, through ST results, to be the diamond
point of Mathematics progress (see the debate on the new axioms to be added
to traditional ST). 

The alternative to this ST is merely to exit out this absolutistic
enterprise for following also the alternative choices, eg; HA, constructive
mathematics, undecidable proofs in both mathematics and computer science,
etc. ; and making use of intuitionist logic (as constructive mathematics
does) and also an organization which is aimed at solving a problem (as was
Cantor' original theory, or Kolmogorov foundation of minimal logic, or
Markov foundation of constructive mathematics, etc.). 

I add that it is illusory to think that a more simple mathematics is ever
reduced by an higher mathematics: a generalization does not prolong a basic
theory without impunity; the costs are the radical variations in meaning of
the common basic notions, eg the notion of limit in all cases reaching the
final point or merely approximating it. These costs cannot be always payed
by more higher abstractions. Goedel's short note (1931a) resuming his
celebrated result, just announced in Königsberg Congress, 5-7 Sept. 1930,
affirms exactly that. 

Best regards
Antonino Drago

-----Messaggio originale-----
Da: fom-bounces at cs.nyu.edu [mailto:fom-bounces at cs.nyu.edu] Per conto di
Harvey Friedman
Inviato: giovedì 9 gennaio 2020 19:10
A: Foundations of Mathematics
Oggetto: [FOM] Improving Set Theory

ZFC has become the standard foundation for mathematics since about
1920. Alternatives have been proposed but not widely endorsed at least
not yet.

I am particularly interested in what people think is lacking or is
flawed about ZFC.

What needs exploring is to what extent and how can ZFC can be
adjusted, modified, improved to meet such objections or requirements.

My general thesis is that ZFC is extremely flexible and supports many
modifications in many different directions for may different purposes,
and that such modifications or adjustments of ZFC are far preferable
to any kind of overhaul in f.o.m.

Harvey Friedman
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