Re: Socialism in SF

"Mike Schilling" <mscottschilling@xxxxxxxxxxx> rote in news:iluxl.14180$8_
3.8436@xxxxxxxxxxxxxxxxxxxx:

Just out of curiosity, how do you take the
Cartesian product of nonempty sets and have the result come out to
be
empty?

How do you, a century after Russell, assume that naive intuition
* accurately describes the universe of sets?


I don't think the axiom of choice is naïve, though making it an axiom is
part of what turns naïve set theory into axiomatic set theory. Nor do I
think Frege's axioms for second order logic are particularly intuitive; in
fact the murkiness of it is why, I think, he got into trouble whereas
Cantor didn't. Frege used a certain axiom to prove Hume's principle, which
is that two sets are the same size if and only if they are in one to one
correspondence. If you toss his (murky) axiom which leads to Russell's
paradox and replace it with the intuitive Hume's principle, you get a
system of second order logic strong enough to prove second order
arithmetic, and no headaches like Russell's paradox. If Frege's axioms had
said in a clear way that you had weird stuff like the set of all sets I
think it would have also been clear that it was not intuitive and
potentially paradoxical.


--
"It's not like there is much that is universal among economists." -- Shawn
Wilson
.

Relevant Pages

  • Re: Uncountable sets
    ... |My intuition says that uncountable sets do not exist. ... axioms except the power set axiom. ... if I give a countable enumeration of nonempty r.e. sets, ...
    (sci.math)
  • Re: Predicativism and natural numbers
    ... > doesn't play much of a role in the acceptance of the existence ... > a set not assumed to satisfy the axiom of induction. ... > induction something we know by "intuition". ... ON THE LINE, or IN THE STRING. ...
    (sci.logic)
  • Re: Socialism in SF
    ... How do you, a century after Russell, assume that naive intuition ... do I think Frege's axioms for second order logic are particularly ... Frege used a certain axiom to prove ... headaches like Russell's paradox. ...
    (rec.arts.sf.written)
  • Re: The Theory of Evolution is a mathematically irrational belief
    ... But Brouwer's intuition led him astray. ... All set theoretic powers are granted by axiom. ... specifying a procedure to do so. ... The need for a constructive procedure is where Brouwer's intuitionism ...
    (talk.origins)
  • Re: Thinking Badly About Infinity
    ... He also uses the Axiom of Intuition: ... Math Level II, as well as scores above 700 in Bio, Spanish and English ... mistake and refuse to admit it, then you are forced to continue the mistake. ...
    (sci.math)