101–110 of 271 posts
Loving this so far! One tiny typo that threw me for a loop for a second was: > when apples are a fruit, then bananas or applies implies bananas or fruits which should be: > when apples are a fruit, then bananas or apples implies bananas or fruits
Will address, thanks!
Pronouns do not have an independent existence. Until the pronoun 'This' resolves to an actual statement which has a valid true/false property, it is a recurrence without termination i.e. a infinite loop, that has no meaning.
Suppose I say: 1. Sky is blue. 2. Previous statement is True. 3. Previous statement is True.
And ask you 'Is Statement 3 is True or False?'. You would say 'First I need to know what 'previous' refers to', so you would do pronoun substitution for the word 'Previous' in Statement 3:
3. 'Previous statement is True' is True.
Since this statement has an unresolved pronoun, you would do pronoun substitution for the word 'Previous' in Statement 2:
3. ''Sky is blue' statement is True' is True.
Since there are no more unresolved pronouns, you would evaluate the statement 'Sky is blue', and then the next Statement 2, and finally Statement 3, and say 'Yes, Statement 3 is True'. So far, so good ?
Now, suppose I say:
1. This statement is False.
And ask you 'Is Statement 1 True or False?'. You would say 'First I need to know what 'This' refers to', so you would do pronoun substitution for the word 'This' in Statement 1:
1. 'This statement is False' statement is False.
Since this statement has an unresolved pronoun ('This'), you would do pronoun substitution again.
- ''This statement is False' statement is False' statement is False.
Since this statement has an unresolved pronoun ('This'), you would do pronoun substitution once more.
- '''This statement is False' statement is False' statement is False' statement is False'
And you will keep doing pronoun substitutions forever since statements with unresolved pronouns cannot have a True/False property.
I find this teenage-girl-at-a-rock-concert-feigned-fascination with self-referential 'paradoxes' silly.
Very off-topic, but long ago when Albert Einstein went for a car ride with Mr. Gödel to become U.S. citizens, Einstein was trying very hard to think of ways to shut him up about a Constitution loophole he discovered. Things didn't go exactly as planned, and in front of the naturalization examiner, Mr. Gödel started blabbing about how he had found a way the U.S could be transformed into a fascist regime... Sources: ht…
The constitution is BS anyway. For example, Article I, section 10 says "No state shall...coin money, emit bills of credit, make any thing but gold and silver a tender in payment of debts..." And look what the US has today; fiat money. No only is it not gold or silver, but it's not event backed by it. Based on this definition, nobody in the US earns any money so nobody should have to pay any income tax since basically…
I think you are confusing the word "state".
Very off-topic, but long ago when Albert Einstein went for a car ride with Mr. Gödel to become U.S. citizens, Einstein was trying very hard to think of ways to shut him up about a Constitution loophole he discovered. Things didn't go exactly as planned, and in front of the naturalization examiner, Mr. Gödel started blabbing about how he had found a way the U.S could be transformed into a fascist regime... Sources: ht…
The constitution is BS anyway. For example, Article I, section 10 says "No state shall...coin money, emit bills of credit, make any thing but gold and silver a tender in payment of debts..." And look what the US has today; fiat money. No only is it not gold or silver, but it's not event backed by it. Based on this definition, nobody in the US earns any money so nobody should have to pay any income tax since basically…
Earlier quoted context omitted.
> It makes it very confusing, especially in light of Godel's other landmark result, i.e Godel's Completeness Theorem which simultaneously applies to many of the same logical systems and can be very very vaguely described as "all true things are provable." If people want to understand how to reconcile this very vague statement with Goedel incompleteness theorem that appears to say that some "true" statements are in fa…
I am trying to follow, but I feel I am hanging Goedel incompleteness theorem -> apply this to "theories" Goedel completeness theorem -> apply this to a specific "model", of the theory So there can be "something true but not provable" in the theory, and "we cannot describe all true things about specific models", but ....what?
Goedel's completeness theorem tells you that for a theory in first-order logic, there's a model that corresponds exactly to that theory: anything that's true in that model can be proven in the theory, and vice versa.
Godel's incompleteness theorem tells you that there's no such model for any system that includes the usual "full" version of arithmetic. A model is a very concrete thing, so any question you ask about the model has an answer - but you don't necessarily know whether that answer is a fact about the theory (i.e. true in all models of the theory - which any statement that's provable would be) or just a fact about that particular model.
It's like if you write some C code, run it, and it prints 4. Does that mean your program always prints 4 according to the C standard? Or did your program do some undefined behaviour, and it just happened that with your particular compiler it printed 4?
Earlier quoted context omitted.
Others have explained why you can't wave away inconsistency (principle of explosion) nor incompleteness (adding new axioms just creates a new axiomatic system with its own Godel sentences). However you might also find it interesting what incompletenesses exist in our own mathematical system (ZFC) -- the most well-known example is the Continuum Hypothesis[1]: > There is no set whose cardinality is strictly between tha…
> being able to prove the CH is an incompleteness in ZFC. It's only incompleteness if the CH is true or false at the semantic level, "outside" of the logic system under discussion. But the CH may be neither true or false, semantically, if the meaning of "existence of a set whose cardinality is strictly between that of the integers and the real numbers" strictly depends on the axioms and logic used to define sets and…
Earlier quoted context omitted.
I am trying to follow, but I feel I am hanging Goedel incompleteness theorem -> apply this to "theories" Goedel completeness theorem -> apply this to a specific "model", of the theory So there can be "something true but not provable" in the theory, and "we cannot describe all true things about specific models", but ....what?
The way I see it is that both theorems are statements about the relationship between a theory and models of that theory. Goedel's completeness theorem tells you that for a theory in first-order logic, there's a model that corresponds exactly to that theory: anything that's true in that model can be proven in the theory, and vice versa. Godel's incompleteness theorem tells you that there's no such model for any system…
Unfortunately it doesn't work if you swap our "for all models" with "there exists a model."
Take the theory of the natural numbers with the new symbol "Special" and the axiom "there exists a number n such that Special(n) holds." In every model of that theory there will be a concrete n that is Special (say 6). However, the theory will never be able to prove that a specific n is Special, precisely because it is under-specified which n it is.
> Godel's incompleteness theorem tells you that there's no such model for any system that includes the usual "full" version of arithmetic.
No in conjunction with Godel's completeness theorem, the incompleteness theorem would say rather that there is no theory that is strictly only the "full" version of arithmetic. Any theory will always be too broad and have too many models that satisfy it, only one of which is the "usual" model of arithmetic.
EDIT: Ah I see I misinterpreted your statement the first time, around, but my objections still hold with modification. Change "the theory of the natural numbers" to a far weaker theory, e.g. the theory of Presburger Arithmetic, and everything else still holds.
Earlier quoted context omitted.
Others have explained why you can't wave away inconsistency (principle of explosion) nor incompleteness (adding new axioms just creates a new axiomatic system with its own Godel sentences). However you might also find it interesting what incompletenesses exist in our own mathematical system (ZFC) -- the most well-known example is the Continuum Hypothesis[1]: > There is no set whose cardinality is strictly between tha…
I thought that independent (undecidable) statements are totally different than the Gödel sentences which demonstrate incompleteness. The latter is a statement which is true in the axiomatic system but which cannot be proven using the axiomatic system. The former is just a statement that essentially has no truth value in the axiomatic system.
I never understood the fascination people have with self-referential 'paradoxes' like : "This statement is False". Pronouns do not have an independent existence. Until the pronoun 'This' resolves to an actual statement which has a valid true/false property, it is a recurrence without termination i.e. a infinite loop, that has no meaning. Suppose I say: 1. Sky is blue. 2. Previous statement is True. 3. Previous statem…
Arithmetic seems to have a number (so to speak) of practical applications as well.
Perhaps it's a bit like theory vs. practice, or maybe mathematics vs. engineering.