[FOM] Status of AC

[joeshipman@aol.com]

Friedman:

>Far far fewer than that, with abbreviation power. Relative 
constructibility
>is not all that bad given my
>http://www.cs.nyu.edu/pipermail/fom/2006-February/009780.html

I was not asking about the length in ZFC "with abbreviation power".

I was asking about the length of your Sigma^1_4 statement in *2nd order 
arithmetic* using a straightforward coding of constructibility (it is 
cheating to use a statement in second-order arithmetic which is not 
obviously equivalent to one which uses straightforward coding, but 
which can be proven to be equivalent by some proof which is long when 
translated into second-order arithmetic).

Are you really saying that the statement which you describe below can 
be expressed in 2nd order arithmetic (the official Z2, without 
abbreviations) in far far fewer where 10,000 symbols, where "expressed 
in" really means that you are straightforwardly translating the 
statement, and not replacing it with a short equivalent statement 
unless the proof *in 2nd order arithmetic* of the equivalence is also 
far far shorter than 10,000 symbols? Because to define "degrees of 
constructibility" in second-order arithmetic (not in ZFC) seems to 
involve a large amount of coding to me.

If it can really be done in 1,000 symbols, that would be worth writing 
down explicitly! (Though it would still be horrible enough to support 
my point that AC is practically irrelevant to mathematics that can be 
formulated in Z2, since your statement, when formulated in Z2, would be 
too long to be of independent interest even at 1,000 symbols).


Friedman's earlier description:

>3. In particular, ZFC proves the negation of the following:
>i. For all n there exists n distinct c-degrees; and
>ii. There is no infinite sequence of distinct c-degrees.

-- JS