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Your method is not a 1:1 mapping of reals to integers, because you can't even define the position of a single irrational number with your method. You don't know

by SamBam 4y ago

Your method is not a 1:1 mapping of reals to integers, because you can't even define the position of a single irrational number with your method. You don't know the final digit, so you don't know the leading digit in your system. You have absolutely no idea the ordering of literally any pair of irrational numbers in your system. > 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.

phkahler 4y ago
>> Your method is not a 1:1 mapping of reals to integers, because you can't even define the position of a single irrational number with your method. You don't know the final digit, so you don't know the leading digit in your system. You have absolutely no idea the ordering of even a single irrational number in your system. 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. ;-)
- SamBam 4y ago
  That did not seem in any way to be your point. You said > 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.
- WastingMyTime89 4y ago
  > Why do we accept that there are real numbers that cannot be defined by a finite string of symbols 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.

by SamBam 4y ago

phkahler 4y ago

>> Your method is not a 1:1 mapping of reals to integers, because you can't even define the position of a single irrational number with your method. You don't know the final digit, so you don't know the leading digit in your system. You have absolutely no idea the ordering of even a single irrational number in your system. 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. ;-)

SamBam 4y ago
That did not seem in any way to be your point. You said > 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.
WastingMyTime89 4y ago
> Why do we accept that there are real numbers that cannot be defined by a finite string of symbols 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.