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Originally Posted by phukraut
I think you may be talking about number theory. I'm not. I'm simply saying that the integers can be described by a set in such a way as there is no confusion about them. We can describe an object such as the set of integers without needing to describe every possible relation of the integers to each other. The former is describing the integers. The latter is number theory.
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Actually, I want to describe the integers and plus and times.
I can't. You can't. You can try, but what you are describing isn't the integers. It contains both things that are the integers and things that are not, under one interpritation.
When you go and say the word 'set', are you talking set theory? If so, you run into the same problem as you do when describing the integers.
Now, you can play fast-and-loose, wave your hands, and pretend their is no confusion. "The integers are {..., -3, -2, -1, 0, 1, 2, 3, ...}", and gloss over the problems. But all GIC is is a set of questions about a description of the integers, and a proof you cannot answer them.
Is 'infinity' in your integers? Are you sure? I'm not.