Using field axioms and order axioms prove the following theorems (explain every step by referencing basic...

Using field axioms and order axioms prove the following theorems (explain every step by referencing basic axioms)

(i) The sets R (real numbers), P (positive numbers) and [1, infinity) are all inductive

(ii) N (set of natural numbers) is inductive. In particular, 1 is a natural number

(iii) If n is a natural number, then n >= 1

(iv) (The induction principle). If M is a subset of N (set of natural numbers) then M = N

The following definitions are given:

A subset S of R is called inductive, if 1 is an element of S and if x + 1 is an element of S whenever x is an element of S.

The intersection of all inductive sets if called the set of natural numbers and is denoted by N

Expert Solution

By the definition of Inductive given here, I have proved the four theorems. Hope you like the solutions. If you have any doubt then please leave a comment about that. I'll try my level best to solve your doubt.and according to my knowledge the fourth is true only if M is inductive. If you think this is wrong please share the picture of the page where this question is written. If you like the solution please give a thumbs up. Thanks in advance.

Colby Messinger answered 2 years ago

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