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

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:

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:

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