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by Corry Shores
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[Griss, “Negationless Intuitionistic Mathematics, I”, entry directory]
[The following is summary. I am not a mathematician, so please consult the original text instead of trusting my summarizations, which are surely mistaken or inelegantly articulated. Bracketed comments and subsection divisions are my own. Proofreading is incomplete, so please forgive my mistakes.]
Summary of
George François Cornelis Griss
(G.F.C. Griss)
“Negationless Intuitionistic Mathematics, I”
1.3
“The Order-Relation”
Brief summary:
(1.3.1) “We define the relation a precedes b, a < b, which has the same meaning as b follows a, b > a, and the relation a immediately precedes b (b immediately follows a).” In this way, any set of terms {1, 2, ..., n} can be arranged in such an order of procession. (1.3.2) If for two numbers in the same ordered set one precedes another, then they are not equal numbers: “If for {1, 2, ..., m} a < b, then a ≠ b.” (1.3.3) Precession is transitive: “Property: If for {1, 2, ..., m} (m > 2) a < b and b < c, then a < c.” (1.3.4) If a number in an ordered set does not equal 1, then it must come after 1: “Property: If a ≠ 1 is an element of {1, 2, ..., m}, then 1 < a.” If a number in an ordered set does not equal the last number, then it must come before it: “Property: If a ≠ m is an element of {1, 2, ..., m}, then a < m.” If b is neither the first nor the last number, then any other number a must either precede or succeed b. “Property: If a and b (b ≠ 1 and b ≠ m) are elements of {1, 2, ..., m}, for each element a that differs from b holds a < b or a > b.” Also, we cannot have negative numbers in sets constructed this way and in accordance with negationless intuitionistic mathematical principles. (1.3.5) If one number a precedes another number b, and if for all the numbers c coming before b, they also come before a, then b immediately follows a: “If a < b and if for each c < b and c ≠ a c < a holds, then b immediately follows a.” Similarly, if a number a precedes another number b, and if for all the other numbers c that come after a and that are not b – if they all come after b, then b immediately follows a (check this quote, as it says b immediately follows b): “If a < b and if for each c > a and c ≠ b c > b holds, then b immediately follows b.” If a number b immediately follows another number a, which itself is not the first number, then for all the other numbers coming before b, if they do not equal a, then they come before a: “If b immediately follows a (a ≠ 1) , then for each c < b and c ≠ a holds c < a.” Similarly, if a number b immediately follows another number a, and b is not the final number, then all the numbers larger than a that are not equal to b would have to come after b: “If b immediately follows a (b ≠ m), then for each c > a and c ≠ b holds c > b.” (1.3.6) Suppose some number b is greater than 1, and it has numbers c that come before it. If some other number a does not equal b and does not equal any of these numbers c coming before b, then a comes after b: “a ≠ b and a ≠ c for each c < b (b ≠ 1) → a > b.” Similarly, suppose some number b is not the last number, and it has numbers c that come after it. If some other number a does not equal b and does not equal any of these numbers c coming after b, then a comes before b: “a ≠ b and a ≠ c for each c > b (b ≠ m) → a < b.” On the basis of these properties, we define the following: “a ≤ b as a = b or a < b and likewise a ≥ b;” “a ≠ c for each c < b (b ≠ 1) → a ≥ b;” “a ≠ c for each c > b (b ≠ m) → a ≤ b;” and “a ≥ 1 and a ≤ m.”
[The Ordering Relation Defined]
[Precession Entails Inequality]
[The Transitivity of Precession]
[Some Ordering Properties]
[Some More Ordering Properties]
[Yet More Ordering Properties]
Summary
[The Ordering Relation Defined]
[“We define the relation a precedes b, a < b, which has the same meaning as b follows a, b > a, and the relation a immediately precedes b (b immediately follows a).” In this way, any set of terms {1, 2, ..., n} can be arranged in such an order of procession.]
[We will define the ordering relation. We have items – numbers – in a set, and we will order them according to which ones precede or follow which other ones; and we will use the less than or greater than symbols. We also more specifically designate when one item immediately proceeds or follows another: “We define the relation a precedes b, a < b, which has the same meaning as b follows a, b > a, and the relation a immediately precedes b (b immediately follows a).” So in {1, 2}, 1 immediately precedes 2. In {1, 2, 3}, 2 immediately precedes 3, and also 1 precedes 3. (There is more to this, see the quotation below.) In this way, we have a series of ordered terms, such that for {1, 2, ..., n} and {1, 2, .... n′}, we have p < n′ for each p of {1, 2, ..., n}. ]
We define the relation a precedes b, a < b, which has the same meaning as b follows a, b > a, and the relation a immediately precedes b (b immediately follows a).
For {1, 2} we have 1 < 2. If a and b are elements of {1, 2} and if a < b, then a = 1 and b = 2. 1 immediately precedes 2. |
For {1, 2, 3} we have 2 < 3 and 1 < 3. If a and b are elements of {1, 2, 3} and if a < b, then b = 3 and a belongs to {1, 2} or a and b belong to {1, 2}. 2 immediately precedes 3.
If, in this way, we have proceeded to {1, 2, ..., n}, for {1, 2, .... n′} we have p < n′ for each p of {1, 2, ..., n}. If a and b are elements of {1, 2, ..., n′} and if a < b, then b = n′ and a belongs to {1, 2, ..., n} or a and b belong to {1, 2, ..., n}. n immediately precedes n′.
(1132-1133)
[Precession Entails Inequality]
[If for two numbers in the same ordered set one precedes another, then they are not equal numbers: “If for {1, 2, ..., m} a < b, then a ≠ b.”]
[The next idea is that if for two numbers in the same ordered set one precedes another, then they are not equal numbers: “If for {1, 2, ..., m} a < b, then a ≠ b.” (See the proof in the quotation below.)]
If for {1, 2, ..., m} a < b, then a ≠ b.
Proof: For {1, 2} the proposition holds. Let the proof have proceeded to {1, 2, ..., n}. If a and b are elements of {1, 2, …, n′} and if a < b, then b = n′ and a belongs to {1, 2, ..., n}, so that a ≠ b or a and b belong to {1, 2, ..., n}, so that a ≠ b.
(1133)
[The Transitivity of Precession]
[Precession is transitive: “Property: If for {1, 2, ..., m} (m > 2) a < b and b < c, then a < c.”]
[ditto]
Property: If for {1, 2, ..., m} (m > 2) a < b and b < c, then a < c.
Proof: {1, 2, 3}: as a < b b belongs to {2, 3} and as b < c b belongs to {1, 2}. So b = 2, a = 1 and c = 3, so that a < c. Let the proof has proceeded to {1, 2, ..., n}. For the elements of {1, 2, ..., n′} is a < b and b < c. c = n′ or c belongs to {1, 2, ..., n}; in both cases b belongs to {1, 2, ..., n}, so a too and a < c.
(1133)
[Some Ordering Properties]
[If a number in an ordered set does not equal 1, then it must come after 1: “Property: If a ≠ 1 is an element of {1, 2, ..., m}, then 1 < a.” If a number in an ordered set does not equal the last number, then it must come before it: “Property: If a ≠ m is an element of {1, 2, ..., m}, then a < m.” If b is neither the first nor the last number, then any other number a must either precede or succeed b. “Property: If a and b (b ≠ 1 and b ≠ m) are elements of {1, 2, ..., m}, for each element a that differs from b holds a < b or a > b.” Also, we cannot have negative numbers in sets constructed this way and in accordance with negationless intuitionistic mathematical principles.]
[ditto]
Property: If a ≠ 1 is an element of {1, 2, ..., m}, then 1 < a. Property: If a ≠ m is an element of {1, 2, ..., m}, then a < m. Property: If a and b (b ≠ 1 and b ≠ m) are elements of {1, 2, ..., m}, for each element a that differs from b holds a < b or a > b.
These properties, just as the following ones, can easily be proved by induction. The condition b ≠ 1 (also b ≠ m) is necessary in the last property, for if b = 1, there is no element a which would satisfy a < b. This cannot be allowed in negationless mathematics (Cf. Introduction). We return to this subject in the theory of sets.
(1133)
[Some More Ordering Properties]
[If one number a precedes another number b, and if for all the numbers c coming before b, they also come before a, then b immediately follows a: “If a < b and if for each c < b and c ≠ a c < a holds, then b immediately follows a.” Similarly, if a number a precedes another number b, and if for all the other numbers c that come after a and that are not b – if they all come after b, then b immediately follows a (check this quote, as it says b immediately follows b): “If a < b and if for each c > a and c ≠ b c > b holds, then b immediately follows b.” If a number b immediately follows another number a, which itself is not the first number, then for all the other numbers coming before b, if they do not equal a, then they come before a: “If b immediately follows a (a ≠ 1) , then for each c < b and c ≠ a holds c < a.” Similarly, if a number b immediately follows another number a, and b is not the final number, then all the numbers larger than a that are not equal to b would have to come after b: “If b immediately follows a (b ≠ m), then for each c > a and c ≠ b holds c > b.”]
[ditto]
If a < b and if for each c < b and c ≠ a c < a holds, then b immediately follows a.
If a < b and if for each c > a and c ≠ b c > b holds, then b immediately follows b.
If b immediately follows a (a ≠ 1) , then for each c < b and c ≠ a holds c < a.
If b immediately follows a (b ≠ m), then for each c > a and c ≠ b holds c > b.
(1133)
[Yet More Ordering Properties]
[Suppose some number b is greater than 1, and it has numbers c that come before it. If some other number a does not equal b and does not equal any of these numbers c coming before b, then a comes after b: “a ≠ b and a ≠ c for each c < b (b ≠ 1) → a > b.” Similarly, suppose some number b is not the last number, and it has numbers c that come after it. If some other number a does not equal b and does not equal any of these numbers c coming after b, then a comes before b: “a ≠ b and a ≠ c for each c > b (b ≠ m) → a < b.” On the basis of these properties, we define the following: “a ≤ b as a = b or a < b and likewise a ≥ b;” “a ≠ c for each c < b (b ≠ 1) → a ≥ b;” “a ≠ c for each c > b (b ≠ m) → a ≤ b;” and “a ≥ 1 and a ≤ m.”]
[ditto]
Finally:
a ≠ b and a ≠ c for each c < b (b ≠ 1) → a > b.
a ≠ b and a ≠ c for each c > b (b ≠ m) → a < b.
From the preceding properties follows, if we define
a ≤ b as a = b or a < b and likewise a ≥ b.
a ≠ c for each c < b (b ≠ 1) → a ≥ b.
a ≠ c for each c > b (b ≠ m) → a ≤ b.
a ≥ 1 and a ≤ m.
Griss, G.F.C. (1946). “Negationless Intuitionistic Mathematics, I,’’ Proceedings of the Koninklijke Nederlandse Akademie van Wetenschappen, 49, 1127–1133.
Journal PDF here:
http://www.dwc.knaw.nl/DL/publications/PU00014659.pdf
Article PDF here:
http://www.dwc.knaw.nl/DL/publications/PU00018278.pdf
Listing of Griss at this journal:
http://www.dwc.knaw.nl/toegangen/digital-library-knaw/?pagetype=publist&search_author=PE00000531
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