Re: the need for relevance

In article <e66d0$478cd55e$82a1e228$12842@xxxxxxxxxxxxxxxx>,
Han de Bruijn <Han.deBruijn@xxxxxxxxxxxxxx> wrote:

Tonico wrote:

For Mr. HdB, the (mathematical, to be sure) fact that there exists a
bijection between the set of natural numbers and the set of even
natural numbers, and thus BY DEFINITION both the set of natural
numbers and the set of even natural numbers have the same cardinality,
is a contradiction.

Why? Because HE KNOWS there are half as many even natural numbers as
there are natural numbers! How does he know that? Who cares! He's said
that anyone knows that, and thus it is so. Period.

No and No. He's said that the actual infinite set of THE naturals does
not exist and that an infinite set of naturals can _only_ be conceived
as a _limiting_ case of a _finite_ set {0,1,2,3,4,5,6,7,8, ... ,n}.
Don't stop the reasoning about this finite set until you're done. Only
at the very end you say: let n become infinite: n -> oo . This is very
standard practice in common calculus.

On the contrary, the much more common calculus practice is to let
epsilon go towards zero, as in derivatives and definite integrals.

IMO it's the only way to approach
infinity, without running the risk of becoming metaphysical. Moreover,
this whole idea is far more comprehensible than all of your _nonsense_
about bijections between a set and a proper subset of itself. What you
actually do is not even _this_. You make _two_ sets, one with naturals
and one with evens, and you say there's a bijection between them. Duh!

Are you saying there isn't a bijection x <--> 2*x ?


There is one and only one real world.

There are at least as many "real worlds" as there are observers, as no
two of them see quite the same world. There is a presumption, quite a
reasonable one, that if one person were to be placed in another's shoes
he would see "the world" as the other sees it. But it is only a
presumption and is quite impossible to prove.



And we _both_ live in it. Whether
you like it or deny it or not.

If the world that I perceive differs from the world you perceive, as it
seems to, how do we adjudicate which is the "real" one?
.

Relevant Pages

  • Re: infinity
    ... > Yes, for your proposed mapping between N and P, ... the largest element would not map to the ... so our bijection never runs into any such endpoint and ... what is the last element in the mapping of naturals to evens? ...
    (sci.math)
  • Re: infinity
    ... In the latter case, there is a bijection, ... >> regarding the evens, and ask how many bits each set's elements has, the evens ... like the bijection between the naturals and the ... > infinite sets, there is no way to use a "running out of" argument, ...
    (sci.math)
  • Re: Uncountable sets in CZF?
    ... >bijection between N and R and thus P? ... numbers in a model V of ZFC, then there is a generic extension V ... first-order logic (or maybe additional axioms to the first-order logic, ... >naturals and the reals. ...
    (sci.math)
  • Re: Cantors circular "proof" that evens = integers
    ... The fact that there was a bijection between the naturals ... The naturals WERE ALREADY THERE ... OUR set theory is 1st-order ZFC and WE KNOW what ... there are no infinite numbers in PA, ...
    (sci.logic)
  • Re: infinity
    ... >>> Are you aware that if I claim I have a bijection from a set ... >>> elements from set A and maps them to elements of set B? ... half does not map to anything in the naturals. ... > do that by REASONING about all of the elements at once. ...
    (sci.math)