Re: fastest way to print a lot of pdf files?

Erik Max Francis <max@xxxxxxxxxxx> wrote:
> Peter T. Breuer wrote:

>> He does. That's WHY he says "show me your bijection". Your "bijection"
>> is a function, and functions are defined by computational procedures (I
>> don't believe, for the sake of this argument, in the existence of
>> functions which are not computable, and I don't believe in existence
>> without a witness to the existence).

> Congratulations, then, you don't believe in mathematics.

No - that's completely false. You are referring to some particular
version of set theory and logic as though it were "mathematics". There
are many such, all _within_ mathematics. Mathematical logic is
precisely the (mathematical) study of such systems (of mathematics and
logic - logic is only a mathematical system whose subject is logic).

In all versions of mathematics, cardinality means the existence of a
bijection. The trick is in "what is a function" (a bijection). If,
within a classic system of mathematics like the one you have in mind,
you imagine a "flatlander" who only believes in functions that
correspond to a computational procedure, you get a model for another
(relatively consistent) version of mathematics (hey, but we know it is
consistent, because it has a model), called "constructivism". Viewed
from within that model, the "laws" look exactly the same as from
outside. In fact, within this system (usually denoted "L"), the axiom
of choice holds - given an infinite set of sets you can choose a
representative from each of them (the infinite set is really a procedure
for creating sets, and a set is really a procedure for enumerating its
elements, so you can simply enumerate the sets and choose the first
member of each). But one can easily construct a slightly bigger model
than L in which there is NO function (that falls within the functions in
the model) that chooses a member from each of a particular sequence of
sets given by a particular function of the intergers (within the model).
It's via reasoning like that and consideration of the relative merits of
such systems that we know that the axiom of choice is unprovable.

One can similarly construct a classic system within a constructivist
one - if we couldn't then we'd know that the relative consistency of
classic set theory was unprovable within constructivism. But proofs ARE
constructions, so that is kind of absurd .. or maybe too close to the
goedel paradox about consistency being unprovable if true :-).


>> Please stop trying to teach sucking eggs to grandmothers. This is a
>> perfectly consistent mathematical view of the universe - the
>> cnstructivist logician's universe. Throwing in the word "mathematics"
>> as though it added something to your armoury won't win you anything.

> Sorry, I didn't realize "mathematical logician" was another way of
> saying "guy who doesn't know what he's talking about." Oh wait, scratch
> that, I did.

No - you didn't. Read up on it. You want to look up books on "model
theory" and "proof theory", as well as elementary books on set theory
that go as far as proving the independence of some axioms by
constructing other models of set theory in which they are and aren't
true (respectively). And you want to look up books on "intuitionist
logic and matehmatics" and "constructivist logic". Please let's not get
down to finitism.


Peter
.

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