[FOM] V = L/crises

[joeshipman at aol.com]

I disagree; the important structures of mathematics, like the real numbers, are isomorphic to structures of well-founded sets, so that any sensible question I can ask about these structures is equivalent to a question about well-founded sets. This is not necessarily true for constructible sets. For example, if you want to know about real-valued measures on the continuum, or whether there is a bijection between countable ordinals and reals, answering these questions "in L" *changes their meaning* in a way that answering the same questions "in V" does not.

-- JS


-----Original Message-----
From: W.Taylor <W.Taylor at math.canterbury.ac.nz>
To: fom <fom at cs.nyu.edu>
Cc: shirtaylor <shirtaylor at hotmail.com>
Sent: Mon, Aug 25, 2014 5:26 pm
Subject: Re: [FOM] V = L/crises


Quoting "Timothy Y. Chow" <tchow at alum.mit.edu>:

> I suspect, though, that the mathematical
> community might be O.K. with V = L even if they didn't *believe* it,

Perhaps this is similar to the way the community is OK with the axiom
of foundation, even though they may not *believe* it, (whatever that means
for each individual).  I have the feeling that many mathematicians would
happily accept an Aczel-like set theory, as it seems fun stuff, but they
know it is "against the rules" for ZFC (as opposed to ZFC-).

Most mathematicians accept foundation because even if there ARE ill-founded
sets, we can ignore them by restricting our attention to the  
well-founded ones.
Similarly, the community might accept V=L because it's handy, &  
philosophically
it's OK because it just restricts attention to the constructible sets,
even if there *may* be other ones.

Bill Taylor

> if they found themselves bumping up against set-theoretical difficulties a
> lot and if V = L promised to provide a "standard" way of dealing with
> them without having to think too much.
>
> Tim


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