4 ms·

The Reals are ordered right? Any two elements can be compared. I'm guessing you might be referring to infinite sets of reals being potentially unordered under

The Reals are ordered right? Any two elements can be compared. I'm guessing you might be referring to infinite sets of reals being potentially unordered under zfc w/o the axiom of choice. In that case, you made up this word 'infinite' so you have to say what it means. I guess calling that a word game is one way to think about it.

raincole 3y ago
> you might be referring to infinite sets of reals being potentially unordered under zfc w/o the axiom of choice. In that case, you made up this word 'infinite' so you have to say what it means. I guess calling that a word game is one way to think about it. Yeah. Since "uncountable infinite" has no real world meaning (maybe in some modern physics it does?), it's hard to say what the natural definition of real numbers is, and things like axiom of choice's true value is quite arbitrary. But even at a less-abstract level, I don't think the comparability of real numbers is so obvious. For example if you just define a (irrational) real number as a non-repeating decimal, or "a program on a Turing machine that prints digits and never halts"[1], then how do we know comp(A, B) halts or not? It's not a proof of that real numbers are not comparable (since it just reduces comp(A,B) to halting problem, not vice versa), but at least for me it's telling that simple things like comparison is not always simple. [1]: Of course it's ill-defined and can't cover all real numbers, since the number of programs on a giving Turing machine is countable.
- enugu 3y ago
  You can encode halting of a program P as a comparison of a computable real number Q with a fixed number R by defining Q as 0.111..1 where each step of P adds one digit of 1 to Q’s expansion. P will halt iff Q is less than R=0.111… Any subset of reals is ordered as it inherits the usual order from reals. The existence of well ordering (related to AOC) is difficult issue). But the trichotomy of A>=<B does fail for a different but useful logic - remove the law of contradiction. There is a number e which is neither equal nor not equal to 0, with e^2=0. This leads to simplifications of concepts and proofs - you can define derivatives without limits for instance. This topic is studied in synthetic differential geometry. But the real response to the comment ‘everything is just a word game’ is ‘just’ is not apt. You are free to fix rules of the game, once done you face questions which are possibly beyond your ability to answer. A person could run a program checking id Fermat equations had solutions in 1950’s. Only In 1990’s we know after great advances (like discovering a route between mountain ranges) that this program wont halt (or ZFC is inconsistent which would be even more surprising).

raincole 3y ago

> you might be referring to infinite sets of reals being potentially unordered under zfc w/o the axiom of choice. In that case, you made up this word 'infinite' so you have to say what it means. I guess calling that a word game is one way to think about it. Yeah. Since "uncountable infinite" has no real world meaning (maybe in some modern physics it does?), it's hard to say what the natural definition of real numbers is, and things like axiom of choice's true value is quite arbitrary. But even at a less-abstract level, I don't think the comparability of real numbers is so obvious. For example if you just define a (irrational) real number as a non-repeating decimal, or "a program on a Turing machine that prints digits and never halts"[1], then how do we know comp(A, B) halts or not? It's not a proof of that real numbers are not comparable (since it just reduces comp(A,B) to halting problem, not vice versa), but at least for me it's telling that simple things like comparison is not always simple. [1]: Of course it's ill-defined and can't cover all real numbers, since the number of programs on a giving Turing machine is countable.

enugu 3y ago
You can encode halting of a program P as a comparison of a computable real number Q with a fixed number R by defining Q as 0.111..1 where each step of P adds one digit of 1 to Q’s expansion. P will halt iff Q is less than R=0.111… Any subset of reals is ordered as it inherits the usual order from reals. The existence of well ordering (related to AOC) is difficult issue). But the trichotomy of A>=<B does fail for a different but useful logic - remove the law of contradiction. There is a number e which is neither equal nor not equal to 0, with e^2=0. This leads to simplifications of concepts and proofs - you can define derivatives without limits for instance. This topic is studied in synthetic differential geometry. But the real response to the comment ‘everything is just a word game’ is ‘just’ is not apt. You are free to fix rules of the game, once done you face questions which are possibly beyond your ability to answer. A person could run a program checking id Fermat equations had solutions in 1950’s. Only In 1990’s we know after great advances (like discovering a route between mountain ranges) that this program wont halt (or ZFC is inconsistent which would be even more surprising).