Re: Socialism in SF

Gene wrote:
"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,

No, but your characterization of it is.

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.

Axiom schema: for any sentrence of first-order logic with one variable
"x", there exists the set {all x such that the sentence is true}.
That's no murkier than ZF's subset axioms, but it leads directly to
Russell's paradox. And it says clearly that there's a set of all
sets: use "x = x"


.

Relevant Pages

  • Re: Socialism in SF
    ... How do you, a century after Russell, assume that naive intuition ... I don't think the axiom of choice is naïve, though making it an axiom is ... system of second order logic strong enough to prove second order ... and no headaches like Russell's paradox. ...
    (rec.arts.sf.written)
  • Godels incompleteness theorm proven wrong
    ... either lead to paradox or end in paradox. ... axiom of reducibility but this axiom was rejected as being invalid by ... Thus just on this point Godel is invalid as by using an axiom most people ... Godel proved that mathematics was inconsistent ...
    (sci.math)
  • Godels incompleteness theorem proven invalid
    ... axiom 1V - axiom of reducibility- in his proof. ... either lead to paradox or end in paradox. ... Thus just on this point Godel is invalid as by using an axiom most people ... Godel proved that mathematics was inconsistent ...
    (sci.math)
  • 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: third infinity !!!
    ... In ordinary set theory, with the axiom of regularity, ... Russell's paradox or ...
    (sci.math)