Every integer has a finite number of numerals. Proof: by definition, each natural number is either 0 or a successor to 0. 0 is finite. And if n is a finite length, then n + 1 is a finite length. So by mathematical induction, all natural numbers have finite length. A similar argument covers the negative numbers, so all integers have finite lengths.
Since the decimal expansion of certain rational numbers are infinite (e.g., 1/3 = 0.333...), the strategy of mapping each number between 0 to 1 to an integer by flipping digits over the decimal point does not work for all rationals, let alone all reals.
> It's position is as precise as your definition.
This wand-waving is actually transforming irrational numbers into rational numbers, and so means you are not actually working with irrational numbers. You can't just say "this works with all irrational numbers, like Pi, so long as we call Pi '3.14'."
Pi is not 3.14, so any mapping system that depends on you cutting off your precision at some finite number of digits is worthless.
That is my point exactly. Why do we accept that there are real numbers that cannot be defined by a finite string of symbols, but there are no integers of that nature.
Getting back to the axiom of choice thing. How do you pick a specific item from one of those ill-defined sets that include numbers that don't have finite descriptions? How do you choose something you can't name? It's absurdity all the way down. ;-)
> I've never been convinced that there are more reals than integers
I'm explaining that your method of mapping reals to integers is flawed, and so does not prove that reals and integers have the same cardinality.
We don't accept it. It's easy to prove that it's the case.
I think you are unfamiliar with how R is defined. The traditional definition of R is a completion of Q with respect to the metric |x-y| (getting all Cauchy sequences to converge). That's larger than infinite sequence of decimals.
Easy!
[sarcasm]
BTW I know rational numbers map to integers, and hence any length tuple can be mapped to integers by pairwise combining them in similar ways. But if any length tuple of integers can map to a single integer, why can't an infinitely long set of integers map to an integer? Like the string of digits in an irrational number represented as a decimal?
Even if they were, pi is not between zero and one, so would not appear in your list.
The rest of your argument is difficult to parse. If you’re interested in this kind of stuff though, I’d really recommend picking some good books on set theory and maybe number theory. Set theory in particular will cover a lot of what you’re talking about.
This sounds similar to an interesting (and often counter-intuitive) way of writing down numbers called 'p-adic numbers' (although that terminology usually requires 'p' to be prime; in your case you'd want the '10-adic' numbers, since you're using base-10)
See e.g. https://en.wikipedia.org/wiki/P-adic_number (here's a video specifically about 10-adic numbers https://www.youtube.com/watch?v=XXRwlo_MHnI )
It's the same argument. If I can accept that there are more reals than reals with finite descriptions, why can't there be integers without finite descriptions/positions?
Because the integers are defined as the finite successors to the number 0 and their additive opposites. See, for instance, the construction section of: https://en.m.wikipedia.org/wiki/Integer
You can start talking about numbers that can't be constructed in this way, but then you've ceased to be talking about the integers.
But it isn't, in any case, an integer.
> Simply flip the digits over the decimal point, so 0.14159 become integer 95141
They key here is that both 0.14159 and 95141 contain a _finite_ number of symbols/digits. There are infinitely many numbers like this, but each of those numbers are themselves finite.
The decimal expansion of pi, OTOH, is _itself_ infinite. The decimal expansion of any irrational number is infinite. There are infinitely many of these things, each of which are themselves infinite.
This is the key thing that causes uncountability. Each of the numbers has infinitely many "parameters" that can be tweaked. If you try to produce an infinite (countable) list that contains all of these infinite numbers, I can always tweak a different parameter for each number in your list and be guaranteed to have produced a number that your list missed.
We can extend this to (I forget the terminology) enumerate the numbers that can be named, by converting the names to integers. Now pi has several location in this mapping, as do a lot of real numbers.
Infinite sets are interesting, but I've never been convinced that there are more reals than integers. It seems plausible but I don't think pi not mapping to a specific integer in the above scheme invalidates the mapping.