Inducing on Q+: Is it Possible?

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Discussion Overview

The discussion revolves around the possibility of using mathematical induction on the set of positive rational numbers (Q+). Participants explore whether induction can be applied in this context and the implications of well-ordering in relation to induction.

Discussion Character

  • Exploratory
  • Technical explanation
  • Debate/contested

Main Points Raised

  • Some participants inquire whether a statement can be proven true for all positive rational numbers using induction, specifically by establishing a base case and certain inductive steps.
  • One participant asserts that it is possible to prove statements for Q+ by induction, suggesting that the method is valid in principle.
  • Another participant mentions the concept of well-ordering, stating that any set that can be well-ordered can be subjected to induction, and highlights the challenges of applying this to rational numbers.
  • There is a reference to the axiom of choice and its implications for well-ordering, indicating that while induction on Q+ is theoretically possible, it may be complex in practice.

Areas of Agreement / Disagreement

Participants express differing views on the feasibility of induction on Q+. While some believe it is possible, others raise concerns about the practicality and complexity of such an approach.

Contextual Notes

The discussion touches on the limitations of induction when applied to rational numbers, particularly regarding the assumptions required for well-ordering and the challenges in establishing a clear inductive framework.

Treadstone 71
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Is it possible to induce on Q+ by showing that a statement is true for n=1 and (n/m=>(n+1)/m AND n/m=>n/(m+1))?
 
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Treadstone 71 said:
Is it possible to induce on Q by showing that a statement is true for n=1 and (n=>n+1 AND n=>n/(n+1))?

First, have you tried induction on the integers?

Positive and Negative.
 
Yes, I have used induction many times before on the integers. My question is whether it is possible prove that a statement is true for all (positive) rational numbers, by induction, in principle.
 
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Treadstone 71 said:
Yes, I have used induction many times before on the integers. My question is whether it is possible prove that a statement is true for all (positive) rational numbers, by induction, in principle.

Yes, that is entirely possible.
 
Excellent, thanks for the reply.
 
Any set that can be well ordered can be inducted upon, and every set can be well ordered (if we accept the axiom of choice), it's just that it's difficult in general, though easier for the rationals since they are lexicographically ordered naturally. The usual way to do it is to assume that there is a set of counter examples, by the well ordering there is a minimal one and we try to deduce a deduction. FOr example one can show that the nCr function is integer valued by induction like this.
 

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