display all the ideas for this combination of texts
11 ideas
10608 | The number of Fs is the 'successor' of the Gs if there is a single F that isn't G [Smith,P] |
Full Idea: The number of Fs is the 'successor' of the number of Gs if there is an object which is an F, and the remaining things that are F but not identical to the object are equinumerous with the Gs. | |
From: Peter Smith (Intro to Gödel's Theorems [2007], 14.1) |
10618 | All numbers are related to zero by the ancestral of the successor relation [Smith,P] |
Full Idea: All numbers are related to zero by the ancestral of the successor relation. | |
From: Peter Smith (Intro to Gödel's Theorems [2007], 23.5) | |
A reaction: The successor relation only ties a number to the previous one, not to the whole series. Ancestrals are a higher level of abstraction. |
10849 | Baby arithmetic covers addition and multiplication, but no general facts about numbers [Smith,P] |
Full Idea: Baby Arithmetic 'knows' the addition of particular numbers and multiplication, but can't express general facts about numbers, because it lacks quantification. It has a constant '0', a function 'S', and functions '+' and 'x', and identity and negation. | |
From: Peter Smith (Intro to Gödel's Theorems [2007], 08.1) |
10850 | Baby Arithmetic is complete, but not very expressive [Smith,P] |
Full Idea: Baby Arithmetic is negation complete, so it can prove every claim (or its negation) that it can express, but it is expressively extremely impoverished. | |
From: Peter Smith (Intro to Gödel's Theorems [2007], 08.3) |
10852 | Robinson Arithmetic (Q) is not negation complete [Smith,P] |
Full Idea: Robinson Arithmetic (Q) is not negation complete | |
From: Peter Smith (Intro to Gödel's Theorems [2007], 08.4) |
10851 | Robinson Arithmetic 'Q' has basic axioms, quantifiers and first-order logic [Smith,P] |
Full Idea: We can beef up Baby Arithmetic into Robinson Arithmetic (referred to as 'Q'), by restoring quantifiers and variables. It has seven generalised axioms, plus standard first-order logic. | |
From: Peter Smith (Intro to Gödel's Theorems [2007], 08.3) |
10068 | Natural numbers have zero, unique successors, unending, no circling back, and no strays [Smith,P] |
Full Idea: The sequence of natural numbers starts from zero, and each number has just one immediate successor; the sequence continues without end, never circling back on itself, and there are no 'stray' numbers, lurking outside the sequence. | |
From: Peter Smith (Intro to Gödel's Theorems [2007], 01.1) | |
A reaction: These are the characteristics of the natural numbers which have to be pinned down by any axiom system, such as Peano's, or any more modern axiomatic structures. We are in the territory of Gödel's theorems. |
10603 | The logic of arithmetic must quantify over properties of numbers to handle induction [Smith,P] |
Full Idea: If the logic of arithmetic doesn't have second-order quantifiers to range over properties of numbers, how can it handle induction? | |
From: Peter Smith (Intro to Gödel's Theorems [2007], 10.1) |
10848 | Multiplication only generates incompleteness if combined with addition and successor [Smith,P] |
Full Idea: Multiplication in itself isn't is intractable. In 1929 Skolem showed a complete theory for a first-order language with multiplication but lacking addition (or successor). Multiplication together with addition and successor produces incompleteness. | |
From: Peter Smith (Intro to Gödel's Theorems [2007], 10.7 n8) |
10604 | Incompleteness results in arithmetic from combining addition and successor with multiplication [Smith,P] |
Full Idea: Putting multiplication together with addition and successor in the language of arithmetic produces incompleteness. | |
From: Peter Smith (Intro to Gödel's Theorems [2007], 10.7) | |
A reaction: His 'Baby Arithmetic' has all three and is complete, but lacks quantification (p.51) |
10067 | Gentzen proved the consistency of arithmetic from assumptions beyond arithmetic [Gentzen, by Musgrave] |
Full Idea: Gentzen proved the consistency of arithmetic from assumptions which transcend arithmetic. | |
From: report of Gerhard Gentzen (works [1938]) by Alan Musgrave - Logicism Revisited §5 | |
A reaction: This does not contradict Gödel's famous result, but reinforces it. The interesting question is what assumptions Gentzen felt he had to make. |