[FOM] Epistemology for new axioms

[martdowd at aol.com]

 FOM: 
 I've posted references to several papers of mine proposing "conservative" new
axioms for set theory.  Following are some thoughts which occur to me in
response to this posting.

I have a suspicion that V=L and weakly compact cardinals do not exist.
This might be called a "tidy" universe, and it is certainly logical that
the universe would be of this character.

I have been arguing since my 2011 paper "Some New Axioms for Set Theory"
(https://urldefense.proofpoint.com/v2/url?u=http-3A__www.ijpam.eu_contents_2011-2D66-2D2_1_1.pdf&d=DwIFaQ&c=slrrB7dE8n7gBJbeO0g-IQ&r=xXZM6ZrkjVxXknjzIxhAvQ&m=VqGEYk4g6HiKTjmct9I17Ro9n_u0BPX2eQrgG3cOxPc&s=MLknnjLnovdn987Xs2ako4UjMeHuNpd7kje-8IYSfdQ&e= ) that the "base" universe
should be "extended", to provide some alleviation of the problems arising
from the non-existence of the "set of all sets", and the handiness of the
notion of the totality of all sets.  Proper classes behave like sets to
some extend.  Inaccessible cardinals provide a mechanism for extending
the base universe to alleviate the situation.  The axioms in the above cited
paper postulate closure properties which have internal coherence and are
logically compelling.  In particular, they can be used to prove the
consisyency of ZFC.

The existence of inaccessible cardinals is generally irrelevant to
"ordinary" mathematics.  Further, it is quite natural to try to eliminate
their use.  Inaccessible cardinals alleviate foundational issues.  It seems
to be a characteristic of the universe that their existence is irrelevant
to ordinary mathematics.
Martin Dowd
-----Original Message-----
From: Timothy Y. Chow <tchow at math.princeton.edu>
To: fom <fom at cs.nyu.edu>
Sent: Thu, Sep 5, 2019 9:17 pm
Subject: Re: [FOM] Epistemology for new axioms

Joe Shipman wrote:

> In fact, for arithmetical statements, I claim the opposite: although 
> there may always be arithmetical statements which some mathematicians 
> will claim to have proven while others will not accept the proofs, there 
> will never be an arithmetical statement such that two groups of 
> mathematicians will permanently disagree about it, with one group 
> claiming it has been proven and the other claiming it has been 
> disproven.

This last statement about arithmetical statements seems very weak.  Do you 
think that there will ever arise a "permanent disagreement" about *any* 
mathematical statement in the sense of one group claiming that it *has 
been proven* and the other group claiming that it *has been disproven*? 
We don't have one group of mathematicians going around saying that V=L 
"has been proven" and another group of mathematicians going around saying 
that V=L "has been disproven."  We don't even have one group of 
mathematicians going around saying that Mochizuki has proven abc and 
another group of mathematicians going around saying that abc has been 
disproven.  A "permanent disagreement" of this type would seem to 
represent a kind of major sea change in the way mathematicians think about 
what the word "proven" means, going well beyond technical distinctions 
between arithmetical statements and non-arithmetical statements.

> ***There is no proposition of any mathematical interest that is known 
> (by ZFC proof) to be relatively consistent with and independent of ZFC, 
> such that mathematicians will eventually have a consensus view that it 
> should be considered as a fundamental axiom with which ZFC should be 
> augmented.***
>
> Here "a consensus view" means comparable to the view that Zermelo set 
> theory should be augmented with the Replacement Axiom to give ZF. In my 
> opinion, statements like CH or V=L or their negations will never be 
> settled because there is no way for a believer to persuade a denier, 
> they will never lead to consequences for which any kind of empirical 
> support can be given such as numerical evidence or necessity for 
> science.

I'm inclined to think that a consensus view, as you've defined it above, 
isn't going to emerge in the forseeable future for *any* statement.  The 
reason is that we now live in an age of mathematical pluralism, where some 
people don't even like infinite sets, let alone ZFC, and some people want 
to use type theory as a foundation, etc.  There's no way to stuff all 
these diverse viewpoints back into Pandora's box and return to the good 
old days when people went along with an axiom set because they could be 
sweet-talked into thinking that they believed those axioms all along 
anyway.

Even something as harmless as the existence inaccessible cardinals seems 
to be stuck with the status of "something we freely assume when we need 
it, but which is flagged explicitly when it is invoked."  If even 
inaccessible cardinals haven't achieved "fundamental axiom" status, then 
what hope do other candidate axioms have?

Tim
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