Okay, my brain isn’t wrapping around this quite properly (though the explanation has already helped me to understand the concepts far better than my college education on the subject has!).
Consider the statement: “There exists no x for which, for any number k, x after k successions is equal to zero.” (¬∃x: ∃k: xS-k-times = 0, k>0 is the closest I can figure to depict it formally). Why doesn’t that axiom eliminate the possibility of any infinite or finite chain that involves a number below zero, and thus eliminate the possibility of the two-sided infinite chain?
Or… is that statement a second-order one, somehow, in which case how so?
Edit: Okay, the gears having turned a bit further, I’d like to add: “For all x, there exists a number k such that 0 after k successions is equal to x.”
That should deal with another possible understanding of that infinite chain. Or is defining k in those axioms the problem?
“For all x, there exists a number k such that 0 after k successions is equal to x” That should deal with another possible understanding of that infinite chain. Or is defining k in those axioms the problem?
I made roughly a similar comment in the Logical Pinpointing post, and Kindly offered a response there.
If I understood him correctly basically it meant “you can’t use numbers to count stuff yet, until you’ve first pinpointed what a number is...”. And repetition isn’t defined in first order logic either.
And while I’m pretty sure you could replace the statement with an infinite number of first-order statements that precisely describe every member of the set (0S = 1, 0SS = 2, 0SSS = 3, etc), you couldn’t say “These are the only members of the set”, thus excluding other chains, without talking about the set—so it’d still be second-order.
It’s a bit worse than that. Even if we defined the “k-successions” operator (which is basically addition), it doesn’t actually let us do what we want. “For all x, there exists a number k such that 0 after k successions is equal to x” is always satisfied by setting k=x, even if x is some weird alternate-universe number like 2*. Granted, I have no clue what “taking 2* successions of 0” means, but...
Okay, my brain isn’t wrapping around this quite properly (though the explanation has already helped me to understand the concepts far better than my college education on the subject has!).
Consider the statement: “There exists no x for which, for any number k, x after k successions is equal to zero.” (¬∃x: ∃k: xS-k-times = 0, k>0 is the closest I can figure to depict it formally). Why doesn’t that axiom eliminate the possibility of any infinite or finite chain that involves a number below zero, and thus eliminate the possibility of the two-sided infinite chain?
Or… is that statement a second-order one, somehow, in which case how so?
Edit: Okay, the gears having turned a bit further, I’d like to add: “For all x, there exists a number k such that 0 after k successions is equal to x.”
That should deal with another possible understanding of that infinite chain. Or is defining k in those axioms the problem?
I made roughly a similar comment in the Logical Pinpointing post, and Kindly offered a response there.
If I understood him correctly basically it meant “you can’t use numbers to count stuff yet, until you’ve first pinpointed what a number is...”. And repetition isn’t defined in first order logic either.
Ah, so the statement is second-order.
And while I’m pretty sure you could replace the statement with an infinite number of first-order statements that precisely describe every member of the set (0S = 1, 0SS = 2, 0SSS = 3, etc), you couldn’t say “These are the only members of the set”, thus excluding other chains, without talking about the set—so it’d still be second-order.
Thanks!
It’s a bit worse than that. Even if we defined the “k-successions” operator (which is basically addition), it doesn’t actually let us do what we want. “For all x, there exists a number k such that 0 after k successions is equal to x” is always satisfied by setting k=x, even if x is some weird alternate-universe number like 2*. Granted, I have no clue what “taking 2* successions of 0” means, but...