Sythe Users: What Do They Know? Do They Know Things?? Lets Find Out! #5

Pretty sure they are incompatible because if not, there could be a bijective function from Z -> R which would make |Z| = |R|

Pretty sure that the 0+ /0- things i was taugh to only use when i wanted to know the behavior of a function as it approaches 0

http://plato.stanford.edu/entries/continuity/

^ Interesting analysis of infinitesimals.

Also good debate. It's more fun being on the answering/public side of these questions as you may have realized.

Edit:

Throughout Cantor's mathematical career he maintained an unwavering, even dogmatic opposition to infinitesimals, attacking the efforts of mathematicians such as du Bois-Reymond and Veronese[38] to formulate rigorous theories of actual infinitesimals. As far as Cantor was concerned, the infinitesimal was beyond the realm of the possible; infiinitesimals were no more than “castles in the air, or rather just nonsense”, to be classed “with circular squares and square circles”. His abhorrence of infinitesimals went so deep as to move him to outright vilification, branding them as “Cholera-bacilli of mathematics.” Cantor's rejection of infinitesimals stemmed from his conviction that his own theory of transfinite ordinal and cardinal numbers exhausted the realm of the numerable, so that no further generalization of the concept of number, in particular any which embraced infinitesimals, was admissible.

Smh. This guy. He should have asked an engineer or a physicist.

 

I'll finish by illustrating an infinite set size paradox that currently exists in set theory:

The interval (0,1] maps 1:1 onto the interval (0,2] via the function f(x) = 2x with no elements left over. Therefore they are the same size.

Split the (0,2] into two sub intervals: (0,1] and (1,2].

The first interval maps exactly onto the second with a function: f(x) = x + 1 with no elements left over. Therefore they are the same size.

If I now take a union of the two intervals (0,1] U (1,2].

The interval (0, 1] maps on to (0,1]U(1,2] with a 1:2 mapping with no elements left over. Therefore the first is half the size of the second.

The only way to get around this it to introduce a new concept: cardinality and replace the concept size with it. Which is what was done. This is called hanging a lantern on your problem. For example a lot of modern astronomy doesn't produce the correct forces or spacing for objects in the universe so they invent a new thing called dark matter. But really this is just hanging a lantern on the problem -- They don't know what the solution to the problem is but by naming it it's no longer regarded as a problem.

You asked before if I support the Axiom of Choice. My answer is no. It took me some time to work it out, but AC necessarily allows the use of infinite sets which in turn introduces various types of infinity as a number (countable, uncountable and possibly more.) It does this without allowing for the logical counterpart of infinitesimals which are completely absent from ZFC set theory. Such a problem this has become that several rigorous attempts have been made to fix ZFC. One example: https://en.wikipedia.org/wiki/Internal_set_theory

Also to return to your original argument:

how can induction proof prove something that is not true?

Alternatively how can a bijective function prove something that is not true?

Your proof relies on assumption of axiom of choice, but mine doesn't. It's not a real axiom (as in a statement which cannot be denied except by accepting that statement). You'd need to prove the axiom of choice before it would stand on level ground with the inductive proof which relies on nothing but standard (in fact a subset of naive) definitions of R, N, Z and set theory.

The reader can decide which wins . And now I've had just about enough formal mathematics reading to last me a life time. gg

 

[Accounts Shop]

I'll finish by illustrating an infinite set size paradox that currently exists in set theory:

The interval (0,1] maps 1:1 onto the interval (0,2] via the function f(x) = 2x with no elements left over. Therefore they are the same size.

Split the (0,2] into two sub intervals: (0,1] and (1,2].

The first interval maps exactly onto the second with a function: f(x) = x + 1 with no elements left over. Therefore they are the same size.

If I now take a union of the two intervals (0,1] U (1,2].

The interval (0, 1] maps on to (0,1]U(1,2] with a 1:2 mapping with no elements left over. Therefore the first is half the size of the second.

The only way to get around this it to introduce a new concept: cardinality and replace the concept size with it. Which is what was done. This is called hanging a lantern on your problem. For example a lot of modern astronomy doesn't produce the correct forces or spacing for objects in the universe so they invent a new thing called dark matter. But really this is just hanging a lantern on the problem -- They don't know what the solution to the problem is but by naming it it's no longer regarded as a problem.

You asked before if I support the Axiom of Choice. My answer is no. It took me some time to work it out, but AC necessarily allows the use of infinite sets which in turn introduces various types of infinity as a number (countable, uncountable and possibly more.) It does this without allowing for the logical counterpart of infinitesimals which are completely absent from ZFC set theory. Such a problem this has become that several rigorous attempts have been made to fix ZFC. One example: https://en.wikipedia.org/wiki/Internal_set_theory

Also to return to your original argument:




Your proof relies on assumption of axiom of choice, but mine doesn't. It's not a real axiom (as in a statement which cannot be denied except by accepting that statement). You'd need to prove the axiom of choice before it would stand on level ground with the inductive proof which relies on nothing but standard (in fact a subset of naive) definitions of R, N, Z and set theory.

The reader can decide which wins . And now I've had just about enough formal mathematics reading to last me a life time. gg

The axiom of choice states that if you have an infinite number of choices to make and they are always the same choice, you are "allowed" to make the assumption up to infinity.

without the acceptance of the axiom of choice, you can't "count" to infinity and thus can't prove the countability of infinite sets

at least, that's what I understand from it

 

[MohtasaUnique]

No no no. This is not how infinitesimals work.

It's an infinitely small number that's still greater than zero. It is the smallest number. Literally every possible (real) number is a multiple of it.

I got it, that's a subtle difference but I understand