Heyting (2.2.3.1) Intuitionism: An Introduction. Section 2.2.3.1, “[Definition of Apartness of Number-Generators]”, 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

 

2.2.3.1

[Definition of Apartness of Number-Generators]

 

 

 

 

 

Brief summary:

(2.2.3.1) We will give a positive definition for inequality (in negationless intuitionistic mathematics), which is apartness. We say that two real number-generators are apart if after some nth term in their series, the succeeding corresponding terms will always be separated by some gap and thus each number-generator is converging upon a different value. Formally:

‘For real number-generators a and b, a lies apart from b, a ⧣ b, means that n and k can be found such that |a_n+p – b _n+p| > 1/k for every p.’

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Contents

 

2.2.3.1

[Apartness (⧣) Defined]

 

Bibliography

 

 

 

 

 

 

Summary

 

2.2.3.1

[Apartness (⧣) Defined]

 

[We will give a positive definition for inequality (in negationless intuitionistic mathematics), which is apartness. We say that two real number-generators are apart if after some nth term in their series, the succeeding corresponding terms will always be separated by some gap and thus each number-generator is converging upon a different value. Formally:

‘For real number-generators a and b, a lies apart from b, a ⧣ b, means that n and k can be found such that |a_n+p – b _n+p| > 1/k for every p.’

(19)]

 

[In the previous section 2.2.2, we discussed the negative notion of equality. In 2.2.2.1 Heyting said:

If a = b is contradictory (that means : if the supposition that a = b leads to a contradiction), we write a ≠ b.

(17, section 2.2.2.1)

And in 2.2.2.2:

If a ≠ b is contradictory, then a = b.

(17, section 2.2.2.2)

This is negative in the sense of providing a disproof by showing a contradiction (but it is not negative in the sense of lacking a proof. See sections 2.2.2.9 and 2.2.2.10). But in negationless intuitionistic mathematics, we prefer positive concepts, even for the inequality of real number-generators. Let us quickly review what a real number-generator is. It is a Cauchy sequence of rational numbers. And a Cauchy sequence is one with a series of rational numbers that progressively tend toward an ultimate value, with the gap between successive numbers narrowing upon that ultimate value. Wildberger, in Math Foundations 111.6, shows this gradual, interchanging convergence of the values with this diagram:

The green line is the value that the series of rationals are tending toward. The idea was that no matter how small an interval you choose, you will be able to find a place in the sequence after which the gaps between successive values (the space above and below the green line) will be less than that arbitrarily small interval. This implies that it is always moving toward some specific value (the green line). In section 2.2.1.2, Heyting gave us a formal definition of a Cauchy sequence.

‘A sequence {a_n} of rational numbers is called a Cauchy sequence, if for every natural number k we can find a natural number n = n(k), such that |a_n+p – a_n| < 1/k for every natural number p.’

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Here, the 1/k is the arbitrarily small interval. The larger the k value, the smaller the interval. The definition here says that no matter how large the k value (and thus no matter how small the interval), there will be some point along the sequence, some n^th term, after which no matter what further point you select (no matter what p), the difference between successive terms will be smaller than that interval. In section 2.2.1.4, we defined a “real number-generator” or just “number-generator” as being such a Cauchy sequence of rational numbers. And in section 2.2.1.6, we defined the coincidence of number-generators:

The number-generators a ≡ {a_n} and b ≡ {b_n} coincide, if for every k we can find n = n(k) such that |a_n+p – b _n+p| < 1/k for every p. This relation is denoted by a = b.

(16)

(In other words, perhaps, although the terms of the two sequences may not be identically the same, if they are coincident, then they still converge upon the same value, and this is because, after a certain point, their corresponding n^th terms will always fall within a gap smaller than any arbitrarily given one. For, it is saying |a_n+p – b _n+p| < 1/k for every p. Now we will define the apartness of two number generators. Let us look at the definition:

For real number-generators a and b, a lies apart from b, a ⧣ b, means that n and k can be found such that |a_n+p – b _n+p| > 1/k for every p [L. E. J. Brouwer 1919A, p. 3].

(19)

Here it is similar to the definition of coincidence, which is symbolized as = and its contradiction as ≠. However, now, the gap between corresponding terms in the two number generators are greater than some interval. So while they may coincide up to some point, after a while, the corresponding n^th terms will always fall outside a gap of some size from one other, no matter how much further you go down the sequence. Thus they will each independently converge at different values. We say that a ⧣ b when they are apart like this. This is a more positive definition, because rather than saying their coincidence or equality is a contradiction, we are saying that they are always separated by a gap.]

But we have already insisted too much on the negative notion of inequality; negative concepts are for us even less important than in classical mathematics; whenever possible we replace them by positive concepts. In the case of inequality between real number-generators we do this by the

Definition :  For real number-generators a and b, a lies apart from b, a ⧣ b, means that n and k can be found such that |a_n+p – b _n+p| > 1/k for every p [L. E. J. Brouwer 1919A, p. 3].

(19)

BROUWER, L. E. J.

1919A. Begründung der mengenlehre unabhängig vom logischen satz vom ausgeschlossenen Dritten. Zweiter Teil. Verhandelingen Akad. Amsterdam 12, N° 7.

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[contents]

 

 

 

 

 

 

 

 

 

Bibliography:

 

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

 

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