Timeline for How do you prove mathematical induction without the notion of a set?
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| Jun 29 at 12:54 | comment | added | lee pappas | Properties of recursively defined functions and sets can often be proved by an induction principle that follows from the recursive definition. For example, the definition of the natural numbers presented here directly implies the principle of mathematical induction for natural numbers: if a property holds of the natural number 0 (or 1), and the property holds of n + 1 whenever it holds of n, then the property holds of all natural numbers (Aczel 1977:742). | |
| Jun 29 at 12:54 | comment | added | lee pappas | @MichaelCarey, axiom P1 gives the answer, and it doesn't use mathematical induction. Let x denote an arbitrary natural number. By P1, X=1 or there is a y such that x=y' and y is a natural number. The principle of induction follows from the recursive definition. | |
| Jun 25 at 21:21 | comment | added | Michael Carey | When you use a recurisive construction on the Natural Numbers, how do you know it works for an arbitrary n? The answer, is that you are implicitly using the recursion theorem ( which uses Mathematical Induction), or you are proving a fact about the recursive construction- also by Mathematical Induction. | |
| Jun 23 at 22:50 | history | edited | lee pappas | CC BY-SA 4.0 |
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| Jun 23 at 5:30 | history | edited | lee pappas | CC BY-SA 4.0 |
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| Jun 23 at 5:23 | history | edited | lee pappas | CC BY-SA 4.0 |
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| Jun 23 at 5:10 | history | edited | lee pappas | CC BY-SA 4.0 |
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| Jun 23 at 4:41 | history | answered | lee pappas | CC BY-SA 4.0 |