# Talk:Trichotomy (mathematics)

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Field:  Foundations, logic, and set theory

## Theology

I added the theological definition of trichotomy to the listing of definitions. meng.benjamin March 27, 2006 16:40 EST

## AC

Removed assertion that Trichotomy is equivalent to AC. As evidenced by the proof at http://en.wikipedia.org/wiki/Cantor%E2%80%93Bernstein%E2%80%93Schroeder_theorem, this isn't true. 129.97.97.134 (talk) 00:30, 11 June 2008 (UTC)

No, Cantor-Bernstein doesn't imply trichotomy. Cantor-Bernstein says that cardinals are partially ordered. Trichotomy says that they are totally ordered. And this is indeed equivalent to AC, because it implies that every set is smaller than some ordinal (and can therefore be well-ordered). So I'm reverting your edit. --Zundark (talk) 16:37, 11 June 2008 (UTC)

## Boolean Expression

If there is a Boolean expression for trichotomy using AND, OR and NOT it would be much appreciated.

The "exclusiveness" of the Trichotomy can be represented using Exclusive OR (XOR). For example: xRy XOR yRx XOR x=y. But if you want to limit yourself only to AND, OR and NOT for some reason. XOR can be represented by some combinations of them as well. Paulmiko (talk) 23:12, 29 June 2011 (UTC)
The Exclusive OR connective (${\displaystyle {\underline {\lor }}}$) cannot be used alone to express strong trichotomy, because the formula
${\displaystyle P\ {\underline {\lor }}\ Q\ {\underline {\lor }}\ R}$ [1]
is true when P, Q, and R are true. Instead, use
${\displaystyle (P\ {\underline {\lor }}\ Q\ {\underline {\lor }}\ R)\land \lnot (P\land Q\land R)}$. [2]
Margaris introduces a ternary logical connective to mean the same thing:
--50.53.53.208 (talk) 20:25, 15 August 2014 (UTC)

## proof?

The sentence "The law of trichotomy was long assumed true without proof; it was proven true at the end of the 19th century.[1]" doesn't make much sence to me, exactely what has been proved?--Sandrobt (talk) 20:47, 11 October 2010 (UTC)

I agree, it doesn't make much sense. I suppose it was meant to say "the trichotomy of the order relation on real numbers was unproven...", still, I am not sure how this is verifiable. So I removed this sentence. ComputScientist (talk) 14:48, 30 January 2011 (UTC)
The sentence that you removed cited "p148 Simon Singh Fermat's Last Theorem". Did you check it? --50.53.47.35 (talk) 18:42, 16 August 2014 (UTC)
Singh has a paragraph on trichotomy, but he does not cite specific sources. He says that "the law of trichotomy states that every number is either negative, positive, or zero" and that "at the end of the last century the law of trichotomy was proved to be true." (Fermat's Enigma, Anchor Books, 1997, pp. 134-5) --50.53.60.86 (talk) 20:32, 3 September 2014 (UTC)
Thanks. I am not sure what or whose proof Singh is referring to. I know that the trichotomy of the ordinals was discussed at the end of the 19th century, but that is different. (NB I am not an expert on the history of maths.) ComputScientist (talk) 10:02, 4 September 2014 (UTC)
OK. Here is another clue. Davis and Hersh say: "When the real numbers are constructed set-theoretically, according to the recipe of Dedekind or Cantor, for example, the law of trichotomy can be proved as a theorem." (p. 373)
The Mathematical Experience, Study Edition
Philip J. Davis, Reuben Hersh, Elena Anne Marchisotto
Springer Science & Business Media, 2011
--50.53.53.132 (talk) 12:34, 6 September 2014 (UTC)
Davis and Hersh appear to be referring to the methods of constructing the real numbers due to Cantor using Cauchy sequences and due to Dedekind using cuts. Mendelson gives details. Although he does not use the term "trichotomy", his Lemma 5.6.7 (p. 225) appears to be the law of trichotomy in terms of Cauchy sequences. He gives a trichotomy law in terms of cuts in Lemma F2 (p. 325). For historical details, he refers to Manheim (p. 191, footnote).
References
--50.53.53.132 (talk) 18:52, 6 September 2014 (UTC)

## Conflict: Trichotomy - Reflexivity

Hello everyone, the Trichotomy focuses on order relations, but: Don't they satisfy the reflexivity per definition? So consider the pairs (x,x), every case is true (xRx, xRx and x=x) and so the "excluding or" gives false.

Best regards -- 138.246.2.199 (talk) 03:55, 7 September 2011 (UTC)

Order relations are usually defined to be reflexive, but they can also be defined to be irreflexive (see Partially ordered set#Strict and non-strict partial orders for details). For trichotomy, it's the irreflexive form that should be used (which is why the article uses < and > rather than ≤ and ≥). --Zundark (talk) 07:56, 7 September 2011 (UTC)

## Strong and Weak Trichotomy

Tarski uses the terms Strong Law of Trichotomy and Weak Law of Trichotomy to describe the two meanings given in the article. He also provides a proof that Strong Trichotomy follows from Weak Trichotomy, Irreflexivity, and Asymmetry. Mendelson provides a similar proof using Weak Trichotomy, Irreflexivity, and Transitivity.

References

--50.53.34.85 (talk) 04:57, 9 August 2014 (UTC)