Behind every human, there is an idea.

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The question is: Why must nonstandard model exist?

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Suppose there is a formula f(x), which holds iff x is standard.

Now, 0 is standard, so f(0). And, if n is standard then n + 1 is too. In formula, f(n) -> f(n + 1). Therefore, we get for all x f(x), by induction.

But, there is a nonstandard model in which PA holds. Therefore in this model f(x) holds for all x, including nonstandard numbers. Contradiction. So no formula defines the concept “standard”.

Of course, we are working in first-order logic, which is quite inexpressive. It doesn’t allow…

PA is equivalent to ZFC – Inf? How?

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The halting pro…

The halting problem is undecidable implies that the idea of halt is not definable. In a precise way.

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The law of competed construction

Thou shall not refer to an object A when defining A. Note that this does not invalidate recursive definitions because in recursive definitions we refer to a smaller object of the same type, not the object under definition itself.

Otherwise it leads to an infinite loop.

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I don’t know whether P vs NP is the right question. But I know that whether P vs NP is independent is the wrong question to ask.

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有的时候不好好珍惜,没有了又想要。

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