Dedekind-Cantor Axiom: What's Needed & Why?

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Discussion Overview

The discussion revolves around the necessity and implications of the Dedekind-Cantor axiom in the context of real numbers and their properties. Participants explore its role in establishing the existence of points on the real line, the relationship between this axiom and other axioms such as the least upper bound axiom, and the implications for the construction of real numbers from rational numbers.

Discussion Character

  • Exploratory
  • Technical explanation
  • Debate/contested

Main Points Raised

  • One participant questions the necessity of the Dedekind-Cantor axiom, suggesting that two binary operations could define a model of real numbers on collinear points in Euclidean geometry.
  • Another participant asserts that the Dedekind axiom is essential for establishing the existence of points on the real line, emphasizing that without it, one might incorrectly assume that the rationals are sufficient to represent the reals.
  • There is a query regarding the equivalence of the least upper bound axiom and the Dedekind axiom, with one participant expressing uncertainty about the least upper bound axiom.
  • A participant defines the least upper bound axiom, explaining that it states every set of real numbers has a least upper bound, and notes that the rationals do not satisfy this axiom.
  • Another participant argues that the Cantor-Dedekind axiom is more of a metamathematical statement about the relationship between geometry and real numbers, rather than a foundational axiom used in constructing the reals.
  • It is noted that the least upper bound axiom is actively used in standard analysis for constructing real numbers from rationals.

Areas of Agreement / Disagreement

Participants express differing views on the necessity and role of the Dedekind-Cantor axiom, with some arguing for its essential nature while others question its necessity. The equivalence of the least upper bound axiom and the Dedekind axiom remains unresolved, with varying levels of understanding among participants.

Contextual Notes

There are limitations in the discussion regarding the definitions and implications of the axioms mentioned, as well as the assumptions made about the relationship between rational and real numbers. The discussion does not resolve the mathematical steps or the implications of the axioms on the construction of real numbers.

Werg22
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I know that the Dedekind-Cantor axiom establishes an isomorphism between the points of any given (extended) Euclidean line. But why is the axiom needed anyway? Can't we define two binary operations on collinear points in Euclidean geometry such that the points of the line taken together with these two operations form a model of the real numbers?
 
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The axiom of Dedekind is an axiom regarding the EXISTENCE of the points on any real line.
All we know form previous algebraic axioms is that 0 is a real numer (neutral element for addition) and that 1 is a real number (neutral element for product).
It is IMPORTANT that 1 \not= 0 and this is in fact another axiom.
These axioms (together with other axioms on sum and products) lead to the existence of a countable multitude of numers in the real line. Namely \mathbb Q.
\mathbb Q itself satisfy every axioms except the Dedekind axiom.
So if you don't insert another axiom you are implicitly assuming that \mathbb Q is a satysfactory system for the real numbers.
That's why we require another axiom.
 
Ah I see. Are the least upper bound axiom and the Dedekind-Axiom essentially equivalent?
 
Werg22 said:
Ah I see. Are the least upper bound axiom and the Dedekind-Axiom essentially equivalent?

I am tempted to say "YES IT IS", but I don't know the least uper bound axiom.
What does it states?
 
If S is a set of real numbers, k is called an upper bound and is a real number if for every s in S s <= k (I'm sure you know this). An upper bound of S, j, is called the least upper bound of S if j <= k for all k's. The axiom states that every set of real numbers possesses a least upper bound. Q fails to satisfy this axiom; suffice to construct a a series composed of rational numbers but converging to an irrational number.
 
The Cantor-Dedekind axiom is not used in the construction of the real numbers. It's more like a metamathematical axiom which says that the "line" of geometry and the "real numbers" are pretty much the same thing. It's more along the lines of Cauchy's delta-epsilon definition of "continuity" and the Church-Turing thesis.

On the other hand, the least-upper-bound axiom is actually used (in standard analysis) to construct the reals from the rationals.
 

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