In mathematics, a [binary] relation (like >, since it considers two natural numbers and then is either true or false, based on which numbers are considered) is just a set of ordered pairs. Within the standard model of the natural numbers, > is just the [infinite] collection of ordered pairs { (2,1) , (3,1) , (3,2) , (4,1) , (4,2) , (4,3) , … }. So, suppose we have two chains of number-thingies...1, 2, 3,… and 1^, 2^, 3^, …. We can make the ‘>’ rule as follows: ” ‘x > y’ if and only if ‘x has a caret and y does not, or (in this case, both numbers must be in the same chain) x is greater than y within its own chain’ ”. This [infinite] collection of ordered pairs would be { (2,1) , (2^,1^) , (1^,1) , (3^,1^) , (3,2) , (3^,2^) , (3,1) , (4^,1^) , (1^,2) , (4^,2^) , (4,3) , (4^,3^) , … }.
So ‘>’ is a valid relation on two disconnected chains of number-thingies, because we define it to be so by fiat. The numbers we’re working with are nonstandard...so there is no reason to expect that there should be some standard, natural meaning for ‘>’.
Important Note: This explanation of ‘>’ does not correspond to a nonstandard model of first-order Peano arithmetic (and, clearly, not the standard model, either). If you want to know more about that, look to earthwormchuck163′s comment. I thought it might be helpful to you to understand it in a case that’s easier to picture, before jumping to the case of a nonstandard model of first-order Peano arithmetic. That case is even more complex than Eliezer revealed within his post. It would probably be extremely helpful to you to learn about well-orders, order types, and the ordinal numbers to get a handle on this stuff. You are more talented than I if you are able to understand it without that background knowledge.
Hope this helps.
Edit: Annoyingly (in this case), the asterisk causes italicization. Changed asterisks to carets.
Edit 2: Changed “operation” to “relation” everywhere, as per paper-machine’s correct comment.
In mathematics, a [binary] operation (like >, since it considers two natural numbers and then is either true or false, based on which numbers are considered) is just a set of ordered pairs.
Not to nitpick, but “>” is a binary relation, not a binary operation.
In mathematics, a [binary] relation (like >, since it considers two natural numbers and then is either true or false, based on which numbers are considered) is just a set of ordered pairs. Within the standard model of the natural numbers, > is just the [infinite] collection of ordered pairs { (2,1) , (3,1) , (3,2) , (4,1) , (4,2) , (4,3) , … }. So, suppose we have two chains of number-thingies...1, 2, 3,… and 1^, 2^, 3^, …. We can make the ‘>’ rule as follows: ” ‘x > y’ if and only if ‘x has a caret and y does not, or (in this case, both numbers must be in the same chain) x is greater than y within its own chain’ ”. This [infinite] collection of ordered pairs would be { (2,1) , (2^,1^) , (1^,1) , (3^,1^) , (3,2) , (3^,2^) , (3,1) , (4^,1^) , (1^,2) , (4^,2^) , (4,3) , (4^,3^) , … }.
So ‘>’ is a valid relation on two disconnected chains of number-thingies, because we define it to be so by fiat. The numbers we’re working with are nonstandard...so there is no reason to expect that there should be some standard, natural meaning for ‘>’.
Important Note: This explanation of ‘>’ does not correspond to a nonstandard model of first-order Peano arithmetic (and, clearly, not the standard model, either). If you want to know more about that, look to earthwormchuck163′s comment. I thought it might be helpful to you to understand it in a case that’s easier to picture, before jumping to the case of a nonstandard model of first-order Peano arithmetic. That case is even more complex than Eliezer revealed within his post. It would probably be extremely helpful to you to learn about well-orders, order types, and the ordinal numbers to get a handle on this stuff. You are more talented than I if you are able to understand it without that background knowledge.
Hope this helps.
Edit: Annoyingly (in this case), the asterisk causes italicization. Changed asterisks to carets.
Edit 2: Changed “operation” to “relation” everywhere, as per paper-machine’s correct comment.
Not to nitpick, but “>” is a binary relation, not a binary operation.
Ha, thanks. I don’t mind nitpicking. I’ll edit the comment.
Actually, a binary relation is a binary operation (it returns 1 if true and 0 if false). You passed up a chance to counter-nitpick the nitpicker.
Yes, if you want a two-sorted theory, then you can make a boolean type and lift all relations to operations.
That’s not the typical use of the word “operation” in model theory, however.
Thanks, that last link was very helpful.