Combining Texts
Ideas for
'Thinking About Mathematics', 'On Formally Undecidable Propositions' and 'The Communist Manifesto'
expand these ideas
|
start again
|
choose
another area for these texts
display all the ideas for this combination of texts
12 ideas
6. Mathematics / B. Foundations for Mathematics / 1. Foundations for Mathematics
8764
|
Categories are the best foundation for mathematics [Shapiro]
|
6. Mathematics / B. Foundations for Mathematics / 4. Axioms for Number / g. Incompleteness of Arithmetic
10072
|
First Incompleteness: arithmetic must always be incomplete [Gödel, by Smith,P]
|
3198
|
Gödel showed that arithmetic is either incomplete or inconsistent [Gödel, by Rey]
|
9590
|
Arithmetical truth cannot be fully and formally derived from axioms and inference rules [Gödel, by Nagel/Newman]
|
11069
|
Gödel's Second says that semantic consequence outruns provability [Gödel, by Hanna]
|
10118
|
First Incompleteness: a decent consistent system is syntactically incomplete [Gödel, by George/Velleman]
|
10122
|
Second Incompleteness: a decent consistent system can't prove its own consistency [Gödel, by George/Velleman]
|
10611
|
There is a sentence which a theory can show is true iff it is unprovable [Gödel, by Smith,P]
|
10867
|
'This system can't prove this statement' makes it unprovable either way [Gödel, by Clegg]
|
6. Mathematics / B. Foundations for Mathematics / 5. Definitions of Number / f. Zermelo numbers
8762
|
Two definitions of 3 in terms of sets disagree over whether 1 is a member of 3 [Shapiro]
|
6. Mathematics / B. Foundations for Mathematics / 7. Mathematical Structuralism / a. Structuralism
8760
|
Numbers do not exist independently; the essence of a number is its relations to other numbers [Shapiro]
|
8761
|
A 'system' is related objects; a 'pattern' or 'structure' abstracts the pure relations from them [Shapiro]
|