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"there is no integer that is not a factor of another integer" I believe this is a tautology. It is true by the definition of integers (which you provide in the

"there is no integer that is not a factor of another integer" I believe this is a tautology. It is true by the definition of integers (which you provide in the statement itself). It therefore doesn't make a statement about non-existence, but rather, the quality of integers.

shkkmo 11y ago
First off, it depends on the assumption that infinity exists. Secondly, just because something is a tautology, does not mean it is not a statement about non-existence.
- Dylan16807 11y ago
  I can think of a simple proof that relies on the ability to always multiply an integer by two, but that's still an integer, not infinity. Why do you need infinity? Also don't forget to say 'nonzero'.
  - shkkmo 11y ago
    > I can think of a simple proof that relies on the ability to always multiply an integer by two, but that's still an integer, not infinity. Why do you need infinity? That proof relies on induction which in turn relies on the existence of infinity. To see how infinity is necessary, assume there are not infinite integers. Let X be the highest positive integer and multiply it by 2, the result does not exist. > Also don't forget to say 'nonzero'. I did forget :), thanks
    - Dylan16807 11y ago
      You don't need a proper 'infinity' to have integers be unbounded. >Let X be the highest positive integer That contradicts the definition of integer, so it doesn't really show anything. Pretty much all I need for the proof is a working successor function, and if you take away the successor function you have something that doesn't even resemble integers.
      - shkkmo 11y ago
        > You don't need a proper 'infinity' to have integers be unbounded. What do you mean by unbounded? If you mean something like a 64 bit integer that loops around to negatives eventually, then the statement either "every (non-zero) integer is a factor of another integer" is no longer true. > That contradicts the definition of integer, so it doesn't really show anything. Which definition does it contradict? > Pretty much all I need for the proof is a working successor function, and if you take away the successor function you have something that doesn't even resemble integers. You can have a successor function, but if you can only apply it a finite number of times if you don't have infinity.
        Dylan16807 11y ago
        > What do you mean by unbounded? There is no largest integer. > If you mean something like a 64 bit integer that loops around to negatives eventually, then the statement either "every (non-zero) integer is a factor of another integer" is no longer true. Sort of, it depends on how you define factor. What stops me from factoring -4 into 4 * 63? > Which definition does it contradict? Where you take counting numbers and add 0 and negatives. If you reach a point where you can't count more, you screwed something up. > You can have a successor function, but if you can only apply it a finite number of times if you don't have infinity. When you give me an integer n, I only need to apply the successor function n times to provide your factoring example. No problem there.
        shkkmo 11y ago
        > Where you take counting numbers and add 0 and negatives. If you reach a point where you can't count more, you screwed something up. Or you ran out of time, or atoms.

shkkmo 11y ago

First off, it depends on the assumption that infinity exists. Secondly, just because something is a tautology, does not mean it is not a statement about non-existence.

Dylan16807 11y ago
I can think of a simple proof that relies on the ability to always multiply an integer by two, but that's still an integer, not infinity. Why do you need infinity? Also don't forget to say 'nonzero'.
- shkkmo 11y ago
  > I can think of a simple proof that relies on the ability to always multiply an integer by two, but that's still an integer, not infinity. Why do you need infinity? That proof relies on induction which in turn relies on the existence of infinity. To see how infinity is necessary, assume there are not infinite integers. Let X be the highest positive integer and multiply it by 2, the result does not exist. > Also don't forget to say 'nonzero'. I did forget :), thanks
  - Dylan16807 11y ago
    You don't need a proper 'infinity' to have integers be unbounded. >Let X be the highest positive integer That contradicts the definition of integer, so it doesn't really show anything. Pretty much all I need for the proof is a working successor function, and if you take away the successor function you have something that doesn't even resemble integers.
    - shkkmo 11y ago
      > You don't need a proper 'infinity' to have integers be unbounded. What do you mean by unbounded? If you mean something like a 64 bit integer that loops around to negatives eventually, then the statement either "every (non-zero) integer is a factor of another integer" is no longer true. > That contradicts the definition of integer, so it doesn't really show anything. Which definition does it contradict? > Pretty much all I need for the proof is a working successor function, and if you take away the successor function you have something that doesn't even resemble integers. You can have a successor function, but if you can only apply it a finite number of times if you don't have infinity.
      - Dylan16807 11y ago
        > What do you mean by unbounded? There is no largest integer. > If you mean something like a 64 bit integer that loops around to negatives eventually, then the statement either "every (non-zero) integer is a factor of another integer" is no longer true. Sort of, it depends on how you define factor. What stops me from factoring -4 into 4 * 63? > Which definition does it contradict? Where you take counting numbers and add 0 and negatives. If you reach a point where you can't count more, you screwed something up. > You can have a successor function, but if you can only apply it a finite number of times if you don't have infinity. When you give me an integer n, I only need to apply the successor function n times to provide your factoring example. No problem there.
        shkkmo 11y ago
        > Where you take counting numbers and add 0 and negatives. If you reach a point where you can't count more, you screwed something up. Or you ran out of time, or atoms.