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The author or concept searched is found in the following 6 entries.
| Disputed term/author/ism | Author |
Entry |
Reference |
|---|---|---|---|
| Infinity Axiom | Field | II 337 Axiom of Infinity/Field: Problem: set theory without axiom of infinity is not "conservative". >Conservativity/Field, >Set theory, >Axioms. |
Field I H. Field Realism, Mathematics and Modality Oxford New York 1989 Field II H. Field Truth and the Absence of Fact Oxford New York 2001 Field III H. Field Science without numbers Princeton New Jersey 1980 Field IV Hartry Field "Realism and Relativism", The Journal of Philosophy, 76 (1982), pp. 553-67 In Theories of Truth, Paul Horwich Aldershot 1994 |
| Infinity Axiom | Quine | IX 205 Def Infinity Axiom/Quine: an infinite number of elements in types should be possible. One possibility is e.g. : Tarski: that there is a non-empty class x², such that each of its elements is a subclass of another element. Russell: for each x² e N ³ there is a class y1 with x² elements: short L² ε N³. (1) Ex² (Ey1(y1ε x²) u ∀y1[y1 ε x² › Ez1(y1 ‹ z1 ε x²)]). Vs: some thought that the question of whether there were infinitely many individuals was more a question of physics or metaphysics. It is inappropriate to let arithmetic depend on it. Russell and Whitehead regretted the infinity axiom and the axiom of choice, and both made special cases dependent on them, as I do most comprehension assumptions. Frege's Natural Numbers/Quine: are plagued by the necessity of infinity axioms, even if we allow type theory, liberalization and cumulative types, or finally heterogeneous classes. Because within each type there is a finite barrier to how large a class can be, unless there are infinitely many individuals. Zermelo's concept of numbers would be a solution here, but brings problems with complete induction. IX 206 Real Numbers/Quine: for them and beyond, however, infinity axioms are always necessary. Infinity Axiom/Zermelo: (5) Ex[Λ ε x u ∀y(y ε x › {y} ε x)]. It postulates a class to which at least all natural numbers in Zermelo's sense belong. It is equivalent to "N ε ϑ" because N is itself an x that satisfies (5), and vice versa, if x satisfies (5), then N < x., and thus "N ε ϑ" according to the exclusion scheme. Unlike Russell's, this infinity axiom says nothing about the existence of individuals. But it separates the last connections to type theory. Zermelo's and Neumann's numbers are even antithetic to cumulative type theory, because such a class breaks the boundaries of all types. Axioms of Infinity/Russell: was caused by the law of subtraction "S'x = S'y > x = y". In other words, it was used so that the natural numbers would not break off. Similarly for the real numbers. But its meaning goes even further: each subsequent type is the class of all subclasses of its predecessor and thus, according to Cantor's theorem, larger than its predecessor. To accept infinitely many individuals therefore means to accept higher infinities without end. For example, the power class in (7) says that {x:x < N} ε ϑ, and this last class is greater than N after the theorem of Cantor. And so it goes further up. Infinity Axiom/Zermelo: breaks the type limits. Quine pro: this frees us from the burden comparable to the type indices, because even in type theory with universal variables we were forced to Frege's version of the natural numbers, which meant recognition of a different 5 in each type (about classes of individuals) of a different 6 in each type, a different N in each type, etc. In addition there is, throughout the whole hierarchy, a multiplication of all details of the theory of real numbers. 3/5 is something different in every following type and also π, Q, R. For all these constants it is practically necessary to keep the type indices. In Zermelo's system with its axiom of infinity such multiplications do not occur with the task of type boundaries. Zermelo's protection was that he avoided classes that were too large. For the reverse assurance that classes cannot exist only if they were larger than all existing classes, very little provision has been made in its segregation scheme. IX 208 Fraenkel and Skolem first did this in their axiom scheme of substitution. II 93 Infinity Axiom/QuineVsRussell: the Principia Mathematica(1) must be supplemented by the axiom of infinity when certain mathematical principles are to be derived. Axiom of infinity: ensures the existence of a class with an infinite number of elements - New Foundations/Quine: instead comes with the universal class of ϑ or x^ (x = x). >Infinity/Quine, >Classes/Quine. 1. Whitehead, A.N. and Russel, B. (1910). Principia Mathematica. Cambridge: Cambridge University Press. |
Quine I W.V.O. Quine Word and Object, Cambridge/MA 1960 German Edition: Wort und Gegenstand Stuttgart 1980 Quine II W.V.O. Quine Theories and Things, Cambridge/MA 1986 German Edition: Theorien und Dinge Frankfurt 1985 Quine III W.V.O. Quine Methods of Logic, 4th edition Cambridge/MA 1982 German Edition: Grundzüge der Logik Frankfurt 1978 Quine V W.V.O. Quine The Roots of Reference, La Salle/Illinois 1974 German Edition: Die Wurzeln der Referenz Frankfurt 1989 Quine VI W.V.O. Quine Pursuit of Truth, Cambridge/MA 1992 German Edition: Unterwegs zur Wahrheit Paderborn 1995 Quine VII W.V.O. Quine From a logical point of view Cambridge, Mass. 1953 Quine VII (a) W. V. A. Quine On what there is In From a Logical Point of View, Cambridge, MA 1953 Quine VII (b) W. V. A. Quine Two dogmas of empiricism In From a Logical Point of View, Cambridge, MA 1953 Quine VII (c) W. V. A. Quine The problem of meaning in linguistics In From a Logical Point of View, Cambridge, MA 1953 Quine VII (d) W. V. A. Quine Identity, ostension and hypostasis In From a Logical Point of View, Cambridge, MA 1953 Quine VII (e) W. V. A. Quine New foundations for mathematical logic In From a Logical Point of View, Cambridge, MA 1953 Quine VII (f) W. V. A. Quine Logic and the reification of universals In From a Logical Point of View, Cambridge, MA 1953 Quine VII (g) W. V. A. Quine Notes on the theory of reference In From a Logical Point of View, Cambridge, MA 1953 Quine VII (h) W. V. A. Quine Reference and modality In From a Logical Point of View, Cambridge, MA 1953 Quine VII (i) W. V. A. Quine Meaning and existential inference In From a Logical Point of View, Cambridge, MA 1953 Quine VIII W.V.O. Quine Designation and Existence, in: The Journal of Philosophy 36 (1939) German Edition: Bezeichnung und Referenz In Zur Philosophie der idealen Sprache, J. Sinnreich (Hg) München 1982 Quine IX W.V.O. Quine Set Theory and its Logic, Cambridge/MA 1963 German Edition: Mengenlehre und ihre Logik Wiesbaden 1967 Quine X W.V.O. Quine The Philosophy of Logic, Cambridge/MA 1970, 1986 German Edition: Philosophie der Logik Bamberg 2005 Quine XII W.V.O. Quine Ontological Relativity and Other Essays, New York 1969 German Edition: Ontologische Relativität Frankfurt 2003 Quine XIII Willard Van Orman Quine Quiddities Cambridge/London 1987 |
| Infinity Axiom | Tarski | Berka I 474 Existence/existence acceptance/Tarski: Problem: if we (...) eliminate the existential conditions in the axioms, so the corresponding allocation disappears. Every expression will continue to correspond with a natural number, but not vice versa to any natural number an expression. >Unambiguity, Berka I 519 Axiom of infinity/Tarski: with him, we renounce the postulate according to which only the right statements in each individual domain should be provable propositions of logic.(1) 1. A.Tarski, Der Wahrheitsbegriff in den formalisierten Sprachen, Commentarii Societatis philosophicae Polonorum. Vol 1, Lemberg 1935 |
Tarski I A. Tarski Logic, Semantics, Metamathematics: Papers from 1923-38 Indianapolis 1983 Berka I Karel Berka Lothar Kreiser Logik Texte Berlin 1983 |
| Infinity Axiom | Wittgenstein | IV 83 Infinity axiom/Russell/Wittgenstein/Tractatus: 5534 would be expressed in the language in that way that there would be infinitely many names with different meanings. >"Not enough names..." Solution: if we avoid illusionary sentences (E.g. "a = a" E.g. "(Ex) x = a") (this cannot be written down in a correct term notation) - then we can avoid the problems with Russell's infinity axiom. >Infinity. |
W II L. Wittgenstein Wittgenstein’s Lectures 1930-32, from the notes of John King and Desmond Lee, Oxford 1980 German Edition: Vorlesungen 1930-35 Frankfurt 1989 W III L. Wittgenstein The Blue and Brown Books (BB), Oxford 1958 German Edition: Das Blaue Buch - Eine Philosophische Betrachtung Frankfurt 1984 W IV L. Wittgenstein Tractatus Logico-Philosophicus (TLP), 1922, C.K. Ogden (trans.), London: Routledge & Kegan Paul. Originally published as “Logisch-Philosophische Abhandlung”, in Annalen der Naturphilosophische, XIV (3/4), 1921. German Edition: Tractatus logico-philosophicus Frankfurt/M 1960 |
| Infinity Axiom | Gödel | Berka I 367 Axiom of infinity/Gödel: the axiom of infinity can be formulated as follows: "There is exactly a countable number of individuals".(1) >Infinity, >Individuals, >Countability, >Quantification, >Validity/Gödel. 1. K. Gödel: Über formal unentscheidbare Sätze der Principia Mathematica und verwandter Systeme I, Mh. Math. Phys. 38 (1931), pp. 175-198. |
Göd II Kurt Gödel Collected Works: Volume II: Publications 1938-1974 Oxford 1990 Berka I Karel Berka Lothar Kreiser Logik Texte Berlin 1983 |
| Infinity Axiom | Hilbert | Berka I 122 Definition Number/logical form/extended function calculus/Hilbert: the general number concept can also be formulated logically: if a predicate-predicate φ (F) should be a number, then φ must satisfy the following conditions: 1. For two equal predicates F and G, φ must be true for both or none of them. 2. If two predicates F and G are not equal in number, φ can only be true for one of the two predicates F and G. Logical form: (F)(G){(φ(F) & φ(G) > Glz (F,G) & [φ(F) & Glz (F,G) > φ(G)]}. The entire expression represents a property of φ. If we designate it with Z (φ), then we can say: A number is a predicate-predicate φ that has the property Z (φ). >Numbers, >Definitions, >Definability, >Infinity, >Axioms, >Axiom systems, >Predicates, >Properties. Problem/infinity axiom/Hilbert: a problem occurs when we ask for the conditions under which two predicate-predicates φ and ψ define the same number with the properties Z (φ) and Z (ψ). Infinity Axiom/equal numbers/Hilbert: the condition for equal numbers or for the fact that two predicate-predicates φ and ψ define the same number is that, that φ(P) and ψ(P) are true for the same predicates P and false for the same predicates. So that the relationship arises: (P)(φ(P) ↔ ψ(P)) I 122 Problem: when the object area is finite, all the numbers are made equal which are higher than the number of objects in the individual domain. >Finiteness/Hilbert, >Finitism, >Finiteness. For example, if a number is e.g. smaller than 10 to the power of 60 and if we take φ and ψ the predicates which define the numbers 10 to the power of 60+1 and 10 high 60 + 1, then both φ and ψ do not apply to any predicate P. The relation (P)(φ(P) ↔ ψ(P)) Is thus satisfied for φ and ψ, that is, φ and ψ would represent the same number. Solution/Hilbert: infinity axiom: one must presuppose the individual domain as infinite. A logical proof of the existence of an infinite totality is, of course, dispensed with(1). 1. D. Hilbert & W. Ackermann: Grundzüge der theoretischen Logik, Berlin, 6. Aufl. Berlin/Göttingen/Heidelberg 1972, §§ 1, 2. |
Berka I Karel Berka Lothar Kreiser Logik Texte Berlin 1983 |
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| Disputed term/author/ism | Author Vs Author |
Entry |
Reference |
|---|---|---|---|
| Frege, G. | Waismann Vs Frege, G. | Waismann I 77 Frege: Definition of the number in two steps a) when two sets are equal. b) Definition of the term "number": it is equal if each element of one set corresponds to one element of the other set. Unique relation. Under Def "Number of a Set"/Frege: he understands the set of all sets equal to it. Example: the number 5 is the totality of all classes of five in the world. VsFrege: how shall we determine that two sets are equal? Apparently by showing such a relation. For example, if you have to distribute spoons on cups, then the relation did not exist before. As long as the spoons were not on the cups, the sets were not equal. However, this does not correspond to the sense in which the word equal is used. So it is about whether you can put the spoons on the cups. But what does "can" mean? I 78 That the same number of copies are available. Not the assignment determines the equivalence, but vice versa. The proposed definition gives a necessary, but not sufficient condition for equal numbers and defines the expression "equal number" too narrowly. Class: List ("school class") logical or term (mammals) empirical. With two lists it is neither emopirical nor logical to say that they can be assigned to each other. Example 1. Are there as many people in this room as in the next room? An experiment provides the answer. 2. Are 3x4 cups equal to 12 spoons? You can answer this by drawing lines, which is not an experiment, but a process in a calculus. According to Frege, two sets are not equal if the relation is not established. You have defined something, but not the term "equal numbered". You can extend the definition by saying that they can be assigned. But again this is not correct. For if the two sets are given by their properties, it always makes sense to assert their "being-assignment", (but this has a different meaning, depending on the criterion by which one recognizes the possibility of assignment: that the two are equal, or that it should make sense to speak of an assignment! In fact, we use the word "equal" according to different criteria: of which Frege emphasizes only one and makes it a paradigm. Example 1. If there are 3 cups and 3 spoons on the table, you can see at a glance how they can be assigned. I 79 2. If the number cannot be overlooked, but it is arranged in a clear form, e.g. square or diamond, the equal numbers are obvious again. 3. The case is different, if we notice something of two pentagons, that they have the same number of diagonals. Here we no longer understand the grouping directly, it is rather a theorem of geometry. 4. Equal numbers with unambiguous assignability 5. The normal criterion of equality of numbers is counting (which must not be understood as the representation of two sets by a relation). WaismannVsFrege: Frege's definition does not reflect this different and flexible use. I 80 This leads to strange consequences: According to Frege, two sets must necessarily be equal or not for logical reasons. For example, suppose the starlit sky: Someone says: "I don't know how many I've seen, but it must have been a certain number". How do I distinguish this statement from "I have seen many stars"? (It is about the number of stars seen, not the number of stars present). If I could go back to the situation, I could recount it. But that is not possible. There is no way to determine the number, and thus the number loses its meaning. For example, you could also see things differently: you can still count a small number of stars, about 5. Here we have a new series of numbers: 1,2,3,4,5, many. This is a series that some primitive peoples really use. It is not at all incomplete, and we are not in possession of a more complete one, but only a more complicated one, beside which the primitive one rightly exists. You can also add and multiply in this row and do so with full rigor. Assuming that the things of the world would float like drops to us, then this series of numbers would be quite appropriate. For example, suppose we should count things that disappear again during counting or others emerge. Such experiences would steer our concept formation in completely different ways. Perhaps words such as "much", "little", etc. would take the place of our number words. I 80/81 VsFrege: his definition misses all that. According to it, two sets are logically necessary and equal in number, without knowledge, or they are not. In the same way, Einstein had argued that two events are simultaneous, independent of observation. But this is not the case, but the sense of a statement is exhausted in the way of its verification (also Dummett) Waismann: So you have to pay attention to the procedure for establishing equality in numbers, and that's much more complicated than Frege said. Frege: second part of the definition of numbers: Def Number/Frege: is a class of classes. ((s) Elsewhere: so not by Frege! FregeVs!). Example: the term "apple lying on the table comes to the number 3". Or: the class of apples lying on the table is an element of class 3. This has the great advantage of evidence: namely that the number is not expressed by things, but by the term. WaismannVsFrege: But does this do justice to the actual use of the number words? Example: in the command "3 apples!" the number word certainly has no other meaning, but after Frege this command can no longer be interpreted according to the same scheme. It does not mean that the class of apples to be fetched is an element of class 3. Because this is a statement, and our language does not know it. WaismannVsFrege: its definition ties the concept of numbers unnecessarily to the subject predicate form of our sentences. In fact, it results the meaning of the word "3" from the way it is used (Wittgenstein). RussellVsFrege: E.g. assuming there were exactly 9 individuals in the world. Then we could define the cardinal numbers from 0 to 9, but the 10, defined as 9+1, would be the zero class. Consequently, the 10 and all subsequent natural numbers will be identical, all = 0. To avoid this, an additional axiom would have to be introduced, the Def "infinity axiom"/Russell: means that there is a type to which infinitely many individuals belong. This is a statement about the world, and the structure of all arithmetic depends essentially on the truth of this axiom. Everyone will now be eager to know if the infinity axiom is true. We must reply: we do not know. It is constructed in a way that it eludes any examination. But then we must admit that its acceptance has no meaning. I 82 Nor does it help that one takes the "axiom of infinity" as a condition of mathematics, because in this way one does not win mathematics as it actually exists: The set of fractions is dense everywhere, but not: The set of fractions is dense everywhere if the infinity axiom applies. That would be an artificial reinterpretation, only conceived to uphold the doctrine that numbers are made up of real classes in the world (VsFrege: but only conditionally, because Frege does not speak of classes in the world). Waismann I 85 The error of logic was that it believed it had firmly underpinned arithmetic. Frege: "The foundation stones, fixed in an eternal ground, are floodable by our thinking, but not movable." WaismannVsFrege: only the expression "justify" the arithmetic gives us a wrong picture, I 86 as if its building were built on basic truths, while she is a calculus that proceeds only from certain determinations, free-floating, like the solar system that rests on nothing. We can only describe arithmetic, i.e. give its rules, not justify them. Waismann I 163 The individual numerical terms form a family. There are family similarities. Question: are they invented or discovered? We reject the notion that the rules follow from the meaning of the signs. Let us look at Frege's arguments. (WaismannVsFrege) II 164 1. Arithmetic can be seen as a game with signs, but then the real meaning of the whole is lost. If I set up calculation rules, did I then communicate the "sense" of the "="? Or just a mechanical instruction to use the sign? But probably the latter. But then the most important thing of arithmetic is lost, the meaning that is expressed in the signs. (VsHilbert) Waismann: Assuming this is the case, why do we not describe the mental process right away? But I will answer with an explanation of the signs and not with a description of my mental state, if one asks me what 1+1 = 2 means. If one says, I know what the sign of equality means, e.g. in addition, square equations, etc. then one has given several answers. The justified core of Frege's critique: if one considers only the formulaic side of arithmetic and disregards the application, one gets a mere game. But what is missing here is not the process of understanding, but interpretation! I 165 For example, if I teach a child not only the formulas but also the translations into the word-language, does it only make mechanical use? Certainly not. 2. Argument: So it is the application that distinguishes arithmetic from a mere game. Frege: "Without a content of thought an application will not be possible either. WaismannVsFrege: Suppose you found a game that looks exactly like arithmetic, but is for pleasure only. Would it not express a thought anymore? Why cannot one make use of a chess position? Because it does not express thoughts. WaismannVsFrege: Let us say you find a game that looks exactly like arithmetic, but is just for fun. Would it notexpress a thought anymore? Chess: it is premature to say that a chess position does not express thoughts. Waismann brings. For example figures stand for troops. But that could just mean that the pieces first have to be turned into signs of something. I 166 Only if one has proved that there is one and only one object of the property, one is entitled to occupy it with the proper name "zero". It is impossible to create zero. A >sign must designate something, otherwise it is only printer's ink. WaismannVsFrege: we do not want to deny or admit the latter. But what is the point of this assertion? It is clear that numbers are not the same as signs we write on paper. They only become what they are through use. But Frege rather means: that the numbers are already there somehow before, that the discovery of the imaginary numbers is similar to that of a distant continent. I 167 Meaning/Frege: in order not to be ink blotches, the characters must have a meaning. And this exists independently of the characters. WaismannVsFrege: the meaning is the use, and what we command. |
Waismann I F. Waismann Einführung in das mathematische Denken Darmstadt 1996 Waismann II F. Waismann Logik, Sprache, Philosophie Stuttgart 1976 |
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