Hi: Given sets A = { 0, 1, 2, ... } and B = { 1, 2, ... }, the
following can be proved:
1. Exists f:B -> A defined by f(x)=x-1 is into and onto A, thus |A|=|
B|.
2. Exists g:B -> A defined by f(x)=x is into but not onto A, thus |A|>|
B|
Given there exists a procedure to convert all (2) to (1) and reversing
the procedure all (1) to (2), why do we say if any bijection exists
between two sets they are equal rather than saying if any injection
but not surjection exists they cannot be equal?

Thanks,
Scott

Hi all,

I was wondering if I could get the benefit of your collective
knowledge: I am looking for a word in philosophy (whether that be of
math/metaphysics/religion) that denotes "everything", as in the whole
continuous plenum that is reality - before we chop it up and partition
it into conceptual objects and categories for everyday use ;)

I'm wondering if there is already a technical (or at least
philosophical) term that describes the whole "fabric" of the universe
of discourse. The closest I have found is something like
"anatta" (thanks to google/wikipedia) but i'm not sure its perfect.
The term "everything" is loaded in that it implies a set of all
things, when really i'm coming from the reverse angle, of one
universal thing that we then cut up as suits whatever task we are
dealing with.

Anyhow, thanks in advance for _any_ guidance you guys can give.

All best, J.

How does one usually go about proving that a formula of the form
forall x exists y phi(x,y)
or
exists x forall y phi(x,y)
isn't equivalent to one of the form
forall x phi(x)
or
exists x phi(x)
where phi(x) doesn't have unbounded quantifiers?

For example, if our language includes a function symbol f. How would
one prove that "forall x exists y f(y)=x" isn't equivalent to some
formula "forall x phi(x)" or "exists x phi(x)" with phi quantifier-
free?

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I have a new preprint that readers of this newsgroup may be interested in.

http://arxiv.org/abs/0805.1385

It concerns Razborov and Rudich's famous paper on "natural proofs," in
which they argue that existing techniques for proving non-relativizing,
non-monotone, superlinear lower bounds in circuit complexity theory all
rely on "natural properties" of Boolean functions, which they demonstrate
cannot be useful for proving polysize circuit lower bounds unless hard
pseudorandom number generators do not exist.

What I can show is that if the definition of "natural" is weakened
slightly, then there provably exist "almost natural properties" that yield
SIZE(n^d) circuit lower bounds on SIZE(n^(d+epsilon)) computable
functions. This result is unconditional. A second result is that, if
assumes that pseudorandom number generators exist, then there exist
"almost-natural proofs" that P != NP. (Of course if pseudorandom number
generators exist, then P != NP trivially; the significance is that there
exist proofs of P != NP having a certain special form.)