[dlkf]

Your critique of the use of ”bigger” in your strawman applies equally to your own use of ”longer” - neither loop will ever terminate. So in what sense is either one longer?

The diagonal proof is arguing that there is no bijection from N to {0, 1}∞ , despite the fact that both are infinite. The sense in which the latter is ”bigger” is that there are always elements left over that are covered by N.

[ludston]

> So in what sense is either one longer?

Neither. That's my point. Literally any definition of something infinite can always be reduced to a procedure that recursively transforms or observes some prior state. To say that one of these functions can produce more distinct states than another is pointless, because the procedure that produces the most states will always be the one that you ran the most times.

There is nothing observably infinite, since it would take infinite time to observe that any given thing was infinite. The only possible proof of infinity would be a machine that runs infinitely quickly. e.g. https://qntm.org/responsibility

[falserum]

> proof of infinity

This combination of words seems strange.

Like: proof of ‘zero’ or proof of ‘left’.

All of them have definitions, not proofs.

(Qualitative distinction of different infinity types has a proof though)

[kbrkbr]

> Literally any definition of something infinite can always be reduced to a procedure that recursively transforms or observes some prior state.

Could you come up with or point to such a procedure for R (the reals)?

As I understand the diagonalization argument you can do that for N, but not for R.

[ludston]

Yes? Take any natural or real number. If it has no decimal point add one. Then append any digit. Repeat as necessary.

[falserum]

1, 1.1, 1.01, 1.001, …

This will never reach 2, so it will not generate all real numbers. (Which was what parent was asking for, to recusively generate all R)

[ludston]

You can define a procedure that reaches 2 very trivially. e.g. you generate all possible 1 digit numbers, then 2 digit numbers, then 3 digit numbers, etc.

This is how natural numbers work in the first place. You're just adding a decimal point to all of the possible places it could go.

[falserum]

Thanks for reply.

When will this reach PI or e or sqrt(2)?

(There are infinitely many numbers that will not be reached by this procedure)

[kbrkbr]

If you could, you could use that procedure to create a mapping

1 => first step in the procedure 2 => second step ...

That in turn would mean that N and R have the same cardinality. This would be news.

[dlkf]

> Literally any definition of something infinite can always be reduced to a procedure that recursively transforms or observes some prior state.

You can’t generate R this way. This is a consequence of Cantor’s proof.

[ludston]

Sure you can. Generate all 1 digit numbers, then generate all 2 digit numbers and so forth. Which number won't be generated?