25 ideas
17749 | Post proved the consistency of propositional logic in 1921 [Walicki] |
Full Idea: A proof of the consistency of propositional logic was given by Emil Post in 1921. | |
From: Michal Walicki (Introduction to Mathematical Logic [2012], History E.2.1) |
17765 | Propositional language can only relate statements as the same or as different [Walicki] |
Full Idea: Propositional language is very rudimentary and has limited powers of expression. The only relation between various statements it can handle is that of identity and difference. As are all the same, but Bs can be different from As. | |
From: Michal Walicki (Introduction to Mathematical Logic [2012], 7 Intro) | |
A reaction: [second sentence a paraphrase] In predicate logic you could represent two statements as being the same except for one element (an object or predicate or relation or quantifier). |
17764 | Boolean connectives are interpreted as functions on the set {1,0} [Walicki] |
Full Idea: Boolean connectives are interpreted as functions on the set {1,0}. | |
From: Michal Walicki (Introduction to Mathematical Logic [2012], 5.1) | |
A reaction: 1 and 0 are normally taken to be true (T) and false (F). Thus the functions output various combinations of true and false, which are truth tables. |
17752 | The empty set is useful for defining sets by properties, when the members are not yet known [Walicki] |
Full Idea: The empty set is mainly a mathematical convenience - defining a set by describing the properties of its members in an involved way, we may not know from the very beginning what its members are. | |
From: Michal Walicki (Introduction to Mathematical Logic [2012], 1.1) |
17753 | The empty set avoids having to take special precautions in case members vanish [Walicki] |
Full Idea: Without the assumption of the empty set, one would often have to take special precautions for the case where a set happened to contain no elements. | |
From: Michal Walicki (Introduction to Mathematical Logic [2012], 1.1) | |
A reaction: Compare the introduction of the concept 'zero', where special precautions are therefore required. ...But other special precautions are needed without zero. Either he pays us, or we pay him, or ...er. Intersecting sets need the empty set. |
17759 | Ordinals play the central role in set theory, providing the model of well-ordering [Walicki] |
Full Idea: Ordinals play the central role in set theory, providing the paradigmatic well-orderings. | |
From: Michal Walicki (Introduction to Mathematical Logic [2012], 2.3) | |
A reaction: When you draw the big V of the iterative hierarchy of sets (built from successive power sets), the ordinals are marked as a single line up the middle, one ordinal for each level. |
17741 | To determine the patterns in logic, one must identify its 'building blocks' [Walicki] |
Full Idea: In order to construct precise and valid patterns of arguments one has to determine their 'building blocks'. One has to identify the basic terms, their kinds and means of combination. | |
From: Michal Walicki (Introduction to Mathematical Logic [2012], History Intro) | |
A reaction: A deceptively simple and important idea. All explanation requires patterns and levels, and it is the idea of building blocks which makes such things possible. It is right at the centre of our grasp of everything. |
17747 | A 'model' of a theory specifies interpreting a language in a domain to make all theorems true [Walicki] |
Full Idea: A specification of a domain of objects, and of the rules for interpreting the symbols of a logical language in this domain such that all the theorems of the logical theory are true is said to be a 'model' of the theory. | |
From: Michal Walicki (Introduction to Mathematical Logic [2012], History E.1.3) | |
A reaction: The basic ideas of this emerged 1915-30, but it needed Tarski's account of truth to really get it going. |
17748 | The L-S Theorem says no theory (even of reals) says more than a natural number theory [Walicki] |
Full Idea: The L-S Theorem is ...a shocking result, since it implies that any consistent formal theory of everything - even about biology, physics, sets or the real numbers - can just as well be understood as being about natural numbers. It says nothing more. | |
From: Michal Walicki (Introduction to Mathematical Logic [2012], History E.2) | |
A reaction: Illuminating. Particularly the point that no theory about the real numbers can say anything more than a theory about the natural numbers. So the natural numbers contain all the truths we can ever express? Eh????? |
17761 | A compact axiomatisation makes it possible to understand a field as a whole [Walicki] |
Full Idea: Having such a compact [axiomatic] presentation of a complicated field [such as Euclid's], makes it possible to relate not only to particular theorems but also to the whole field as such. | |
From: Michal Walicki (Introduction to Mathematical Logic [2012], 4.1) |
17763 | Axiomatic systems are purely syntactic, and do not presuppose any interpretation [Walicki] |
Full Idea: Axiomatic systems, their primitive terms and proofs, are purely syntactic, that is, do not presuppose any interpretation. ...[142] They never address the world directly, but address a possible semantic model which formally represents the world. | |
From: Michal Walicki (Introduction to Mathematical Logic [2012], 4.1) |
17755 | Ordinals are the empty set, union with the singleton, and any arbitrary union of ordinals [Walicki] |
Full Idea: The collection of ordinals is defined inductively: Basis: the empty set is an ordinal; Ind: for an ordinal x, the union with its singleton is also an ordinal; and any arbitrary (possibly infinite) union of ordinals is an ordinal. | |
From: Michal Walicki (Introduction to Mathematical Logic [2012], 2.3) | |
A reaction: [symbolism translated into English] Walicki says they are called 'ordinal numbers', but are in fact a set. |
17756 | The union of finite ordinals is the first 'limit ordinal'; 2ω is the second... [Walicki] |
Full Idea: We can form infinite ordinals by taking unions of ordinals. We can thus form 'limit ordinals', which have no immediate predecessor. ω is the first (the union of all finite ordinals), ω + ω = sω is second, 3ω the third.... | |
From: Michal Walicki (Introduction to Mathematical Logic [2012], 2.3) |
17757 | Members of ordinals are ordinals, and also subsets of ordinals [Walicki] |
Full Idea: Every member of an ordinal is itself an ordinal, and every ordinal is a transitive set (its members are also its subsets; a member of a member of an ordinal is also a member of the ordinal). | |
From: Michal Walicki (Introduction to Mathematical Logic [2012], 2.3) |
17758 | Ordinals are transitive sets of transitive sets; or transitive sets totally ordered by inclusion [Walicki] |
Full Idea: An ordinal can be defined as a transitive set of transitive sets, or else, as a transitive set totally ordered by set inclusion. | |
From: Michal Walicki (Introduction to Mathematical Logic [2012], 2.3) |
17760 | Two infinite ordinals can represent a single infinite cardinal [Walicki] |
Full Idea: There may be several ordinals for the same cardinality. ...Two ordinals can represent different ways of well-ordering the same number (aleph-0) of elements. | |
From: Michal Walicki (Introduction to Mathematical Logic [2012], 2.3) | |
A reaction: This only applies to infinite ordinals and cardinals. For the finite, the two coincide. In infinite arithmetic the rules are different. |
17762 | In non-Euclidean geometry, all Euclidean theorems are valid that avoid the fifth postulate [Walicki] |
Full Idea: Since non-Euclidean geometry preserves all Euclid's postulates except the fifth one, all the theorems derived without the use of the fifth postulate remain valid. | |
From: Michal Walicki (Introduction to Mathematical Logic [2012], 4.1) |
13949 | All models of Peano axioms are isomorphic, so the models all seem equally good for natural numbers [Cartwright,R on Peano] |
Full Idea: Peano's axioms are categorical (any two models are isomorphic). Some conclude that the concept of natural number is adequately represented by them, but we cannot identify natural numbers with one rather than another of the isomorphic models. | |
From: comment on Giuseppe Peano (Principles of Arithmetic, by a new method [1889], 11) by Richard Cartwright - Propositions 11 | |
A reaction: This is a striking anticipation of Benacerraf's famous point about different set theory accounts of numbers, where all models seem to work equally well. Cartwright is saying that others have pointed this out. |
18113 | PA concerns any entities which satisfy the axioms [Peano, by Bostock] |
Full Idea: Peano Arithmetic is about any system of entities that satisfies the Peano axioms. | |
From: report of Giuseppe Peano (Principles of Arithmetic, by a new method [1889], 6.3) by David Bostock - Philosophy of Mathematics 6.3 | |
A reaction: This doesn't sound like numbers in the fullest sense, since those should facilitate counting objects. '3' should mean that number of rose petals, and not just a position in a well-ordered series. |
17634 | Peano axioms not only support arithmetic, but are also fairly obvious [Peano, by Russell] |
Full Idea: Peano's premises are recommended not only by the fact that arithmetic follows from them, but also by their inherent obviousness. | |
From: report of Giuseppe Peano (Principles of Arithmetic, by a new method [1889], p.276) by Bertrand Russell - Regressive Method for Premises in Mathematics p.276 |
17754 | Inductive proof depends on the choice of the ordering [Walicki] |
Full Idea: Inductive proof is not guaranteed to work in all cases and, particularly, it depends heavily on the choice of the ordering. | |
From: Michal Walicki (Introduction to Mathematical Logic [2012], 2.1.1) | |
A reaction: There has to be an well-founded ordering for inductive proofs to be possible. |
15653 | We can add Reflexion Principles to Peano Arithmetic, which assert its consistency or soundness [Halbach on Peano] |
Full Idea: Peano Arithmetic cannot derive its own consistency from within itself. But it can be strengthened by adding this consistency statement or by stronger axioms (particularly ones partially expressing soundness). These are known as Reflexion Principles. | |
From: comment on Giuseppe Peano (Principles of Arithmetic, by a new method [1889], 1.2) by Volker Halbach - Axiomatic Theories of Truth (2005 ver) 1.2 |
17635 | Arithmetic can have even simpler logical premises than the Peano Axioms [Russell on Peano] |
Full Idea: Peano's premises are not the ultimate logical premises of arithmetic. Simpler premises and simpler primitive ideas are to be had by carrying our analysis on into symbolic logic. | |
From: comment on Giuseppe Peano (Principles of Arithmetic, by a new method [1889], p.276) by Bertrand Russell - Regressive Method for Premises in Mathematics p.276 |
17742 | Scotus based modality on semantic consistency, instead of on what the future could allow [Walicki] |
Full Idea: The link between time and modality was severed by Duns Scotus, who proposed a notion of possibility based purely on the notion of semantic consistency. 'Possible' means for him logically possible, that is, not involving contradiction. | |
From: Michal Walicki (Introduction to Mathematical Logic [2012], History B.4) |
1556 | By nature people are close to one another, but culture drives them apart [Hippias] |
Full Idea: I regard you all as relatives - by nature, not by convention. By nature like is akin to like, but convention is a tyrant over humankind and often constrains people to act contrary to nature. | |
From: Hippias (fragments/reports [c.430 BCE]), quoted by Plato - Protagoras 337c8 |