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by Corry Shores
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[Mathematics, Calculus, Geometry, Entry Directory]
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[Heyting’s Intuitionism: An Introduction, entry directory]
[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.
Arithmetic
2.2
“Real Number Generators”
2.2.1
Definition; Relation of Coexistence
Brief summary:
(2.2.1.1) We will examine the theory of real numbers in intuitionistic mathematics by beginning with Cantor’s theory. (2.2.1.2) 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. Formally:
‘A sequence {an} 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 |an+p – an| < 1/k for every natural number p.’
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(2.2.1.3) We can devise an example using a sequence, namely the decimal series of π, and make a stipulation regarding some part of it, even though we may not even know if such a part of it does in fact exist. [This perhaps shows us an instance where we cannot effectively determine n(k).] (2.2.1.4) We call a Cauchy sequence of rational numbers a “real number-generator,” or just simply a “number-generator,” if that leads to no confusion. (2.2.1.5) “Two number-generators a ≡ {an} and b ≡ {bn} are identical, if an = bn for every n. We express this relation by a ≡ b.” [This perhaps means that if each nth term in both series is equal to the other, then the number-generators are identical.] (2.2.1.6) The second definition is: “The number-generators a ≡ {an} and b ≡ {bn} coincide, if for every k we can find n = n(k) such that |an+p – b n+p| < 1/k for every p. This relation is denoted by a = b.” [This perhaps is to say that although the terms of the two sequences may not be identically the same, they still converge upon the same value.] (2.2.1.7) There is a theorem about coinciding number-generators, namely, that they are reflexive, symmetrical, and transitive. (2.2.1.8) Heyting remarks: “Given any number-generator a ≡ {an}, a number generator b ≡ {bn} can be found such that a = b and that the sequence {bn} converges as rapidly as we wish. For instance, in order that |bn+p – bn| < 1/n for every n and p, it suffices to take bk = an(k) for every k.” [Perhaps the idea is that for every number-generator, we can find another coinciding one, with the identical one being one option.] (2.2.1.9) We can abbreviate a number generator v = {vn} as just v, and vn (without curly brackets) would be the nth component in the sequence v. (2.2.1.10) We will define real numbers in chapter 3, after dealing with set theory, which is requisite.
[Introducing the Topic]
[Cauchy Sequences Defined]
[An Example]
[Definition 1a: The Real Number-Generator]
[Definition 1b: The Identity of Number Generators]
[Definition 2: The Coincidence of Number-Generators]
[A Theorem on Coinciding Number-Generators: Reflexivity, Symmetry, Transitivity]
[Remark: Finding Coinciding Number-Generators]
[Abbreviation for Number Generators]
[Postponing Real Numbers Until After Set Theory]
Summary
[Introducing the Topic]
[We will examine the theory of real numbers in intuitionistic mathematics by beginning with Cantor’s theory.]
[ditto]
INT. Yes, but at the next station, that of real numbers, we enter a totally different landscape. As in the classical mathematics, so in intuitionism different equivalent theories of real numbers are possible [L. E. J. Brouwer 1919A, p. 3; A. Heyting 1935]. I shall briefly expound Cantor’s theory, which has some advantages for our purpose.
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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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HEYTING, A.
1935. Intuitionistische wiskunde. Mathematica B (Leiden) 4, p. 72–82, 123–136; 5, p. 62–80, 105–112; 7, p. 129-141.
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[Cauchy Sequences Defined]
[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. Formally:
‘A sequence {an} 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 |an+p – an| < 1/k for every natural number p.’
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[(It gets technical, and I will need to build from more basic ideas that we have addressed previously. Let me first give the quotation, and we will break it down as best as I can, but you are advised to consult a real mathematician here.
Let us suppose that the theory of rationals, including their order relations, has been developed. A sequence {an} 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 |an+p – an| < 1/k for every natural number p. This must be so understood, that, given k, we are able to determine effectively n(k).
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And we begin with the first line:
Let us suppose that the theory of rationals, including their order relations, has been developed.
So we are assuming that we have an idea of rational numbers (which are ones that can be expressed as an integer over an integer. See Wildberger, Math Foundations 13, 01.30.] And we also have an idea of their ordering, which may be the idea of sequence that we get next.
A sequence {an} 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 |an+p – an| < 1/k for every natural number p.
First we need to understand what a “sequence {an}” is. I am not entirely certain, but it would seem to be, at this early stage, a series of increasing rational numbers. (It may be something like the series Wildberger calculates in Math Foundations 92, 12.25.) Such sequences can have any sort of series on increasing integers, but we will talk about a particular kind, called Cauchy sequences. We discussed them already in Wildberger, Math Foundations 111, sections 111.1-111.8, and I will try to convert that as best as I can to what is given here. The basic nature of the Cauchy sequence is that it is an increasing series of rational numbers, but as you go down along the sequence, they tend toward a certain value progressively and interchangingly. So the first one will be very high for instance from the ultimate value, then next will be very low (but still closer), soon one will be a little high, the next a little low. And presumably they converge upon a 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). So let us go very slowly here through the definition.
A sequence {an} of rational numbers is called a Cauchy sequence, if for every natural number k ...
The new concept here is the natural number k. But as we will see, it will function ultimately as the arbitrarily small value that we mentioned above. We obtain it by dividing k by one, as we learn soon enough. So the idea is that no matter how large you make k (and ultimately, no matter how small an interval you choose), the sequence will continue by narrowing between that gap, starting from some place within the sequence. Let us continue:
A sequence {an} of rational numbers is called a Cauchy sequence, if for every natural number k we can find a natural number n = n(k),
I am not entirely certain, but it seem that this n is like the N in Wildberger’s diagram:
It signifies the nth item in the sequence. Maybe n = n(k) is like a function saying that to the k value there corresponds a place along the sequence that fulfills the stipulations to come. But I am not sure. Yet, if that were the case, then the formula would make sense for me. To continue:
A sequence {an} 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 |an+p – an| < 1/k for every natural number p.
Let us first notice the 1/k. This we said was like the arbitrarily small number. We also now have p. It will serve the function of saying that no matter where you go after that nth term in sequence, the differences between any two successors will be less than that arbitrarily small value. Let us break that down:
|an+p – an|
In the first place, we are finding an absolute value. It is like the gap or distance between two successive values. Suppose p is 1. That means we are dealing with the n value and its successor. The gap between them will be less than the arbitrary value.
|an+p – an| < 1/k
Now, since in a Cauchy sequence the gap narrows, that means any gap further down the sequence will have to be smaller than that first one and thus smaller than the arbitrarily small value. So no matter how large the p (that is, no matter how far along the series you choose to go after that point), the gap between successive values will fit within the arbitrarily small interval.)]
Let us suppose that the theory of rationals, including their order relations, has been developed. A sequence {an} 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 |an+p – an| < 1/k for every natural number p. This must be so understood, that, given k, we are able to determine effectively n(k).
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[An Example]
[We can devise an example using a sequence, namely the decimal series of π, and make a stipulation regarding some part of it, even though we may not even know if such a part of it does in fact exist. [This perhaps shows us an instance where we cannot effectively determine n(k).]]
[(I do not comprehend this example, so please consult the quotation below. I am guessing wildly that the example is to illustrate the previous sentence, “This must be so understood, that, given k, we are able to determine effectively n(k),” by showing a case where the corresponding n term cannot be found, at least currently. Sorry, please see for yourself.)]
Example. The sequence a ≡ {2–n} is a Cauchy sequence. Let the sequence b ≡ {bn} be defined as follows : If the nth digit after the decimal point in the decimal expansion of π is the 9 of the first sequence 0123456789 in this expansion, bn = 1, in every other case bn = 2–n. b differs from a in at most one term, so b is classically a Cauchy sequence, but as long as we do not know whether a sequence 0123456789 occurs in π, we are not able to find n such that |bn+p – bn| < 1/2 for every p; we have no right to assert that b is a Cauchy sequence in our sense.
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[Definition 1a: The Real Number-Generator]
[We call a Cauchy sequence of rational numbers a “real number-generator,” or just simply a “number-generator,” if that leads to no confusion.]
[ditto]
Definition 1. A Cauchy sequence of rational numbers is a real number-generator. Where no confusion is possible, we shall speak briefly of a number-generator.
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[Definition 1b: The Identity of Number Generators]
[“Two number-generators a ≡ {an} and b ≡ {bn} are identical, if an = bn for every n. We express this relation by a ≡ b.” [This perhaps means that if each nth term in both series is equal to the other, then the number-generators are identical.]]
[I am not certain, but the next idea I am supposing is the following. We will now define the identity between number generators. We will say if each nth term in each series (so the first item in the first sequence, along with the first item in the other, then the second item in the first sequence, along with the second item in the other, etc.) are the same for both sequences, then they are identical, which we write as using ≡: “]
Two number-generators a ≡ {an} and b ≡ {bn} are identical, if an = bn for every n. We express this relation by a ≡ b. The following notion of coincidence is more important.
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[Definition 2: The Coincidence of Number-Generators]
[The second definition is: “The number-generators a ≡ {an} and b ≡ {bn} coincide, if for every k we can find n = n(k) such that |an+p – b n+p| < 1/k for every p. This relation is denoted by a = b.” [This perhaps is to say that although the terms of the two sequences may not be identically the same, they still converge upon the same value.]]
[I do not understand the following definition for the coincidence of number-generators, but I wonder, guessingly, if it is the following. In the identity of generators, each nth term needed to be equal for each sequence. But perhaps now we will have two sequences that may not have the same series of terms, but still converge upon the same value. Yet it is probably something else. There also seems to be the idea that, after a certain point, their corresponding nth terms will always fall within a gap smaller than any arbitrarily given one. So maybe the idea is that eventually the pairings will coincide even if they did not begin that way. I am guessing wildly.)]
Definition 2. The number-generators a ≡ {an} and b ≡ {bn} coincide, if for every k we can find n = n(k) such that |an+p – b n+p| < 1/k for every p. This relation is denoted by a = b.
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[A Theorem on Coinciding Number-Generators: Reflexivity, Symmetry, Transitivity]
[There is a theorem about coinciding number-generators, namely, that they are reflexive, symmetrical, and transitive.]
[ditto (For the meanings of these terms, see for instance Graham Priest’s Introduction to Non-Classical Logic sections 21.8.2 and 21.8.3, or section 9.2.6.)]
Theorem. The relation of coincidence between number-generators is reflexive, symmetrical and transitive. The easy proof is well-known.
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[Remark: Finding Coinciding Number-Generators]
[Heyting remarks: “Given any number-generator a ≡ {an}, a number generator b ≡ {bn} can be found such that a = b and that the sequence {bn} converges as rapidly as we wish. For instance, in order that |bn+p – bn| < 1/n for every n and p, it suffices to take bk = an(k) for every k.” [Perhaps the idea is that for every number-generator, we can find another coinciding one, with the identical one being one option.]]
[(I do not get the next point, but I will guess it is the following. For any number-generator, we can find another one that coincides with the first, with one possibility being an identical one. See the quotation please.)]
Remark. Given any number-generator a ≡ {an}, a number | generator b ≡ {bn} can be found such that a = b and that the sequence {bn} converges as rapidly as we wish. For instance, in order that |bn+p – bn| < 1/n for every n and p, it suffices to take bk = an(k) for every k.
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[Abbreviation for Number Generators]
[We can abbreviate a number generator v = {vn} as just v, and vn (without curly brackets) would be the nth component in the sequence v.]
[ditto]
If, in the following, a number-generator is denoted by one letter, v say, it will be silently understood that it can also be denoted by {vn}, so that vn is the nth component of the sequence v.
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[Postponing Real Numbers Until After Set Theory]
[We will define real numbers in chapter 3, after dealing with set theory, which is requisite.]
[ditto]
As the notion of a real number presupposes the fundamental notions of set theory, I postpone the definition of a real number (as a set of coincident number-generators) till chapter III.
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Heyting, Arend. Intuitionism. An Introduction. Amsterdam: North-Holland, 1956.
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