Heyting (2.2.3.8) Intuitionism: An Introduction. Section 2.2.3.8, “[The Contradiction of Apartness as Coincidence or Equality]”, summary

 

by Corry Shores

 

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[The following is summary. I am not a mathematician, so please consult the original text instead of trusting my summarizations, which are possibly mistaken and probably inelegantly articulated. Bracketed comments and subsection divisions are my own. Proofreading is incomplete, so please forgive my mistakes.]

 

 

 

 

Summary of

 

Arend Heyting

 

Intuitionism: An Introduction

 

2.2.3

Apartness-Relation Between Number-Generators

 

2.2.3.8

[The Contradiction of Apartness as Coincidence or Equality]

 

 

 

 

Brief summary:

(2.2.3.8) The third theorem: If two number-generators can be proven to not be apart, then they are coincident or equal:

If a ⧣ b is impossible, a = b.

 

 

 

 

 

 

Contents

 

2.2.3.8

[The Contradiction of Apartness as Coincidence or Equality]

 

Bibliography

 

 

 

 

 

 

Summary

 

 

 

2.2.3.8

[The Contradiction of Apartness as Coincidence or Equality]

 

[The third theorem: If two number-generators can be proven to not be apart, then they are coincident or equal:

If a ⧣ b is impossible, a = b ]

 

[We are dealing with real number-generators (see section 2.2.3.1). And in that section 2.2.3.1, we said that they are apart (⧣) if somewhere along their series there will always be a gap between their corresponding terms and thus that they will converge upon different values. Now we get the theorem that if we can disprove that two number-generators are apart, then we can conclude they are coincident and equal (see section 2.2.1.6): If a ⧣ b is impossible, a = b.]

Theorem 3.   If a ⧣ b is impossible, a = b [L. E. J. Brouwer 1925, p. 254].

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BROUWER, L. E. J.

1925. Intuitionistische Zerlegung mathematischer Grundbegriffe. Jahresbericht deutsch. Math. Ver. 33, p. 251–256.

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[contents]

 

 

 

 

 

 

 

 

 

Bibliography:

 

Heyting, Arend. Intuitionism. An Introduction. Amsterdam: North-Holland, 1956.

 

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