Empty Set doesn't exist

If you cant sense it, then it doesnt exist.

You cant see emptiness, therefore emptiness doesnt exist.

>>79
you are defective.
You mean I'm not "ashkenazi"? I'm perfectly okay with that.

77>>

Is that already defined?
(if not, i dub thee 'negative subset logic [type 0 architecture]' ^^ )

[Top-Set] - probably should always have an 'Empty' negative set
so, ts_A = +{1,2,3,4,..,7} / -{} (like all normal sets?)

Then another top set, which is a (datawise) subset of A can be defined just as ts_B = +{1,2} / -{}

[Subset] ts_A . ts_B = +{1,2} / -{3,4,..,7}

>>81
No. I mean you were born with flawed logic. One with more imagination is greater that one without the capacity to imagine.

>>80
It's easy to get contradictions or paradoxes in natural language - just look at the liar paradox, or various self-refernece sentences.

Contradictions and paradoxes are not always obvious, even when talking about formal systems. For example, naive set theory is inconsistent. We don't even know if modern set theories are consistent. By Godel's incompleteness theorms, we cannot even know if arithmetic is consistent, and that any consistent system containing (or being stronger than) arithmetic cannot ever claim its own consistency as a theorem (if you do, the system is inconsistent). Claiming consistency as an axiom is okay(not that you'd really know if the new system is consistent), but any belief about arithmetic's consistency is a matter of "religion" (since it cannot be known, despite that it would be utterly strange for it to not be consistent) - you cannot know that in a countable infinitity of inferences (in well-defined first order logic) there is no contradiction. Still, such belief is common in almost all scientists, along with belief such as any finite sentence written in the language of First Order Logic + Peano Arithmetic has a truth value (true or false). In more complex systems, such a belief may not be warranted - some statements can be undecidable or independent in the system, this is especially common with infinitary set theories, examples of independent axioms: Continuum hypothesis, Axiom of Choice, various "high" non-constructive cardinal axioms.

You define something, but you don't know if it's sound or consistent(free of contradictions), or if it's true or false.
Let me give a more complex example, consider Goldbach's conjecture "Every even integer greater than 2 can be expressed as the sum of two primes". It can be defined in arithmetic. It can be computably verified (given unbounded resources) by a Turing Machine, yet if it's true, the process will never terminate (it will never find an integer which isn't the sum of 2 primes). Which means that to prove or disprove Goldbach's conjecture (and many other mathematical hypotheses) one must show if one particular process halts or not.
Some set theories makes bets about the behavior of the infinite, or in this case, infinite processes, and if those bets are correct, you will be able to say if some process terminates or not, even if you would not know of that given only the "beliefs" of Peano Arithmetic (definitions of successor, addition, multiplication and an induction axiom; if you don't like the induction axiom, you can avoid it, and you'll get Robinson Arithmetic instead, but in RA, even if you can still compute anything, you can prove much less, not even simple sentences like: "x + y = y + x").
 
Consider Goodstein's theorem ( http://en.wikipedia.org/wiki/Goodstein's_theorem ), it states that some particular (computable) sequence terminates, yet it cannot be proven in Peano Arithmetic, but can be proven in some stronger systems (such as some set theories). Of course, we do know that the Halting problem isn't generally solvable, but the idea here that with stronger systems, more things can be proven and more (particular) cases of the problem can be solved. The downside of such betting is that the stronger you make the theory (by adding axioms) the more you risk adding contradictions and making it inconsistent (can prove anything, thus it no longer talks about truth).

Mathematicians cannot always know if a theory has contradictions or not ("always" knowing will require you to have a solution to the halting problem), despite knowing its axioms, but they can hope to discover such contradictions if they exist (otherwise they will work on a false theory).

>>83
One with more imagination is greater that one without the capacity to imagine.
How "greater"?

>>84
By Godel's incompleteness theorms, we cannot even know if arithmetic is consistent
As I said "there are no contradictions, except those we invent/define."

Mathematicians are trying to solve nonsense problems, they created themselves by some wrong definition of "consistency" in a poorly defined framework, that rests on some crazy axioms (see ZFC), purpose of which a layman wont get without tracing full 20st century history of mathematics, which is full of hacks and silly conventions.

You define something, but you don't know if it's sound or consistent(free of contradictions), or if it's true or false.
It's consitent, when it's parts are consistent, together with their composition.

they can hope to discover such contradictions if they exist (otherwise they will work on a false theory).
Like the Banach-Tarski Theorem, which postulates that given single orange you can transform it into two oranges, by the sole power of applying Set Theory axioms. Behold The Wonder of Infinity's Creation!

>>86
Mathematicians are trying to solve nonsense problems, they created themselves by some wrong definition of "consistency" in a poorly defined framework, that rests on some crazy axioms (see ZFC), purpose of which a layman wont get without tracing full 20st century history of mathematics, which is full of hacks and silly conventions.
I'm sorry, but to me consistency makes perfect sense, all these terms are well-defined. It makes sense within computer science, first order logic and the various arithmetical theories.
I do partially agree that the syntax of some parts of math is a bit messy and it could be done in a more easily parsable manner, but this doesn't change the fact that the semantics would still stay the same even if you chose a better syntax (look at some theorem provers if you want examples of more "parsable" syntaxes).
I'm not an expert on Set Theory, but as far as I understand it, originally there was the simpler naive set theory, with rather intuitive axioms, but it proved inconsistent (see Russell's paradox), so they had to choose some axioms that limit what sets are and how they can be defined. The axioms themselves may seem a bit strange at first as some of them do appear non-constructive. If that bothers you, you could look at a constructive or iterative approach of building sets, one example that I like is described in the first chapters of Boolos' "Logic, Logic, Logic" - it takes a few simple axioms which are intuitive enough (as opposed to the ones in ZFC which make you wonder where they appeared from) and then it shows how sets are built in it and also shows that most ZF axioms are theorems in that particular theory! You might not like it because it has infinity in it, but it can't be avoided, even with only natural numbers: the process of listing the numbers obtain from 0 and a successor function never terminates - the list/set is infinite (countably). Instead, just understand what infinity is and where it comes from and then you can even look at things like ordinals which are well-defined (constructively even, and a computer/theorem prover can talk about them too, despite being 'infinite'!).
It's consitent, when it's parts are consistent, together with their composition.
How do you know the parts are consistent?
Consistency in math means that starting with some axioms and some logic (inference rules), you will never prove any syntactically valid statements in that language to be BOTH true and false. It's fairly simple and well-defined. You can even write a program that could find an error in some inconsistent axiomatic system (of course, for most modern ones, this is highly unlikely and your theorem prover which is searching for an inconsistency will never halt if the theory is consistent, but you'll never know this!).
Like the Banach-Tarski Theorem, which postulates that given single orange you can transform it into two oranges, by the sole power of applying Set Theory axioms. Behold The Wonder of Infinity's Creation!
I don't find that paradox that much stranger than saying that the cardinality of some continous real interval like [0,1] has the same as the cardinality of all real numbers, or that there are as many even natural numbers as natural numbers or as rational numbers, yet there are more reals than naturals. Infinity is counterintuitive like that, but if you can show a bijective function between 2 sets, then their cardinality is the same. It may be strange, but as I said before, there is no reason why you should set a higher bound for natural numbers, and if you do, you're only inviting trouble for your theories (I don't even know of a single consistent ultrafinitist theory, but maybe you know more about them than me). Not that anyone forces you to work with infinity directly, computable functions only work with finite numbers, just that one has to acknowledge that there's a countable infinity of them (both computable functions and natural numbers).

>>78
Set A is a subset of A, it just isn't a proper subset.

Fuck all of you.
>>40

>>86
Like the Banach-Tarski Theorem, which postulates that given single orange you can transform it into two oranges, by the sole power of applying Set Theory axioms. Behold The Wonder of Infinity's Creation!
This is what Jesus used to feed all those people.

>>68
Like the Banach-Tarski Theorem, which postulates that given single orange you can transform it into two oranges, by the sole power of applying Set Theory axioms. Behold The Wonder of Infinity's Creation!
Also worth noting that those apples are nothing like our apples. Our apples are made of finite atoms occupying finite space and that you can't really divide them too much (up to subatomic particles, although thinking of them like that might be wrong). On the other hand, a "set theory" sphere can be cut with infinitely precise detail, that is, it would take infinite bits to specify where you want to cut (that's a real for you) and how. This obviously wouldn't work with a finitely detailed apple (or one which only works with rationals, which still allow for unbounded "detail").

tl;dr: Set theoretical universe sphere != physical sphere made of atoms.

>>90
What makes you think, you cant create the atoms or subatomic particles themself out of nothing? If you believe in N+1, then nothing would stop you from believing in creation.

>>91
That would depend on the particulars of the laws of physics. Someone existing within some particular mathematical structure cannot actually change it, as they are themselves part of the structure. Of course, if whatever laws of physics allow for such an act, sure, it just doesn't seem like our own have any such possibilities (although technically, there's the quantum foam).

A belief in computability (thus natural numbers) in the ontology just means that there's a lot of possibilities as far as laws of physics go, as well as an inevitable first-person indeterminacy regarding in which particular state (or universe, if you want to use that term) you happen to be. It even tells you how you can change your "viewpoint" to one which you might like more (read that novel I mentioned for an example). However, getting one particular mathematical structure to be something else is of course impossible, for example: given the the usual definitions of Peano Arithmetic within the standard interpretation of arithmetic, it's impossible for 1+1=3 to be true (if PA is consistent). For some other system with very different definitions and semantics of 1,3,+,= or what one means by truth, that sentence may very well be true.
Belief in some countable infinity doesn't mean that suddenly the world breaks or that you can do impossible things.

>>87
I'm sorry, but to me..
Everything is subjective.

originally there was the simpler naive set theory, with rather intuitive axioms
In which way they were "intuitive"? How can the property of being "unordered" be intuitive?

it proved inconsistent (see Russell's paradox)
Usingly a nonsensical construction, based on unrestricted quantifier "all".

The axioms themselves may seem a bit strange at first as some of them do appear non-constructive.
They are just hacks, whose sole purpose is to patch their naive theory and hide problems under the carpet.

How do you know the parts are consistent?
Using senses? Decomposition should always terminate into sensible physical terms.

I don't find that paradox that much stranger than saying that the cardinality of some continous real interval like [0,1] has the same as the cardinality of all real numbers
both are nonsense.

>>92
That would depend on the particulars of the laws of physics.
Some time ago, "laws" supported your ideal continuous space.
http://en.wikipedia.org/wiki/Newtonian_physics

>>92
Belief in some countable infinity doesn't mean that suddenly the world breaks or that you can do impossible things.
Just like the belief in any deity wont make it suddenly appear. Still, religious fanatics are a danger to society.

>>95
What does religion have to do with maths?

>>96

Cantor's obsession with mathematical infinity and God's transcendence eventually landed him in an insane asylum.

For the Hindu math genius Ramanujan an equation "had no meaning unless it expresses a thought of God."

The prolific Hungarian mathematician Paul Erdos imagined a heavenly book in which God has inscribed the most elegant and yet unknown mathematical proofs.

If a `religion' is defined to be a system of ideas that contains unprovable statements, then Godel taught us that mathematics is not only a religion, it is the only religion that can prove itself to be one. -- John D. Barrow, Between Inner Space and Outer Space, Oxford University Press, 1999, p 88.

Suppose we loosely define a religion as any discipline whose foundations rest on an element of faith, irrespective of any element of reason which may be present. [Atheism], for example, would be a religion under this definition. But mathematics would hold the unique position of being the only branch of theology possessing a rigorous demonstration of the fact that it should be so classified. -- H. Eves, Mathematical Circles, Boston: Prindle, Weber and Schmidt, 1969.

The mind of man being finite, when it treats of things which partake of infinity, it is not to be wondered at if it run into absurdities and contradictions, out of which it is impossible it should ever extricate itself, it being of the nature of infinite not to be comprehended by that which is finite. -- George Berkeley

It's what I call "mental masturbation", when you engage is some pointless intellectual exercise that has no possible meaning. -- Linus Torvalds

>>93
Everything is subjective.
Not entirely sure what that's supposed to mean, but if you insist...
In which way they were "intuitive"? How can the property of being "unordered" be intuitive?
Unordered just means you don't care about ordering. Imagine you have a list of items and you only test if an item is in the list and don't care about the order of the items on the list.
They are just hacks, whose sole purpose is to patch their naive theory and hide problems under the carpet.
I'm not so sure. The axioms of ZFs were a bit unnatural, but the iterative version does make sense to me.
Using senses? Decomposition should always terminate into sensible physical terms.
For a subjective idealist, you sure use the term 'physical' a lot. Physical is just an indexical property, and saying that "only this" exists is a stronger and more complex belief than "this" exists. Some would even argue that using that name is not too different from calling it magic.
>>94
I'm not talking about whatever current laws of physics we have found by induction. I'm talking about whatever the universal law is (which we might have a chance of discovering at the end of our inductive process), regardless of our knowledge of it - we can observe that the world behaves by precise laws, even if we don't know all their exact specifics (for now).
Newtonian physics is correct given the right context, but not correct in more extended contextss.
Same goes for quantum mechanics and general relativity. The 2 of them aren't even compatible in their current form and finding a way to reconcile them is a challenge for theoretical physics.
>>95
Just like the belief in any deity wont make it suddenly appear.
Of course. However, nobody is having you work with infinity directly, only that you at least not place a particular bound on naturals, otherwise you are severly limited about what you can talk about or think about.

Still, religious fanatics are a danger to society.
Popular religions are silly and some of them can lead to bad behavior. However, any belief which cannot be directly proven is of "religious" nature. There are many such beliefs which can be justified, even if not directly proven, in the end, you have to bet on some things being true or false, otherwise you can't even talk about anything at all, or make any choices. You do know that a system without any axioms, or one where all the valid sentences are theorems is an inconsistent one? It cannot talk about anything at all and can prove any falsity. You have to draw a line somewhere and have some initial assumptions - that's what "religion" actually is. You're free to use your best reasoning and meta techniques to pick whatever theory has the best chance of being true - that's what science is about. Saying that "religion" is bad is only because most people have very nonsensical and irrational religious beliefs. To many regular people the term just means anti-epistemology, where they will gobble up anything they hear as truth, but that's not what I'm talking about. The moment you realize that you must hold some (at least tentative) beliefs to be able to do anything or reason about anything, that's mostly "religion", even if those beliefs are perfectly rational. I could even argue that one should be very careful about what beliefs they have and understand how they test, verify or falsify them.

>>98
Unordered just means you don't care about ordering. Imagine you have a list of items and you only test if an item is in the list and don't care about the order of the items on the list.
Algorithm have to check doubles them in some order.

CHECK MY CENTO-DUBS

>>97
If a `religion' is defined to be a system of ideas that contains unprovable statements, then Godel taught us that mathematics is not only a religion, it is the only religion that can prove itself to be one. -- John D. Barrow, Between Inner Space and Outer Space, Oxford University Press, 1999, p 88.
That much I agree with, however it is a religion that almost all scientists and mathematicians subscribe to, wether they do it consciously or not. Without it, we can hardly get anywhere. I have no problem with "religious" beliefs as long as they are probably true. This doesn't mean that all of them are - if you dislike set theory and think it's false, you're welcome to find inconsistencies in it, surely you won't be the first ultrafinitist to try!

>>98
For a subjective idealist, you sure use the term 'physical' a lot. Physical is just an indexical property, and saying that "only this" exists is a stronger and more complex belief than "this" exists. Some would even argue that using that name is not too different from calling it magic.
I use it as a synonym to sensible, the same way I use term "reality" to refer to the things I can sense. To change one's world, one have to change language.

Here is an interesting article (scientology ideas were proven to work on practice):
http://www.xenu.net/archive/books/us/us-08.html

>>102
Oh, I see, so you're using a different definition that I am.
Not that I'm not partially guilty of that too, the meaning of the term "religious" belief that I mostly used was in the sense of provably unprovable statements which nonetheless can posses a truth value (false or true).

>>98
A list can impose an order. The canonical example is a linked list.

>>104
Sure it can, but you could only choose to define APIs on it which never take into account the ordering. In which case, the list is de-facto unordered (as long as its internals are not exposed).

>>104
And more to the point, this linked list, which imposes an order, can be a series of unsorted items.

>>105
What the hell are you talking about you idiot? You don't *define* an API it.

>>107
*on it.*

>>103
In English, the word "agreement" can have many meanings, such as the act of agreeing, an understanding, or a contract; in Scientology the word means the agreement of two or more people about reality, which is said to exist only when there is agreement that it exists.

>>109
http://en.wikipedia.org/wiki/Consensus_reality

>>89
Praise the lord for he created ZFC.

>>104
linked list
linked is shit

>>104
This is your daily reminder that the statement every subset of a countable set is also countable is absolutely true and never fails under any condition.

Keep denying the truth you mental midget, I'm sure it will change some day. Oh wait...

>>113
Here we go again.

>>114
Well the person who wrote

Unordered just means you don't care about ordering. Imagine you have a list of items and you only test if an item is in the list and don't care about the order of the items on the list.


is totally clueless about the concept of a 'list'.

>>115
Actually, now that I think about it, the person who wrote

>Unordered just means you don't care about ordering. Imagine you have a list of items and you only test if an item is in the list and don't care about the order of the items on the list.

ts clueless about unordering. I can cite several trivial programming examples where an unordered list gets rearranged when performing a delete() operation.

List is ship

>>115-116
I was talking about what the concept 'unordered' means in math. Of course, all memory is ordered on a computer, and memory itself corresponds to some particular circuit with a particular structure and so on. Unordered in programming means that there is a "contract" where you don't use the order, or merely don't expose the order through some interface - the internals of how you hold the items is irrelevant. Unordered in math is just that, but without looking at the implementation (if you do, you're talking about a particular interpretation of some concept).
I wanted this discussion to be civil, so I refrained from resort to restating your past history as >>113 did, but it's hard to do so when you keep confusing interfaces with implementations, and the "what should be" with "what is".

>>118
Of course, all memory is ordered on a computer, and memory itself corresponds to some particular circuit with a particular structure and so on.

I never said nor implied anything about the memory you halfwit. I was talking about the value(stored) in that memory.

>Unordered in programming means that there is a "contract" where you don't use the order, or merely don't expose the order through some interface - the internals of how you hold the items is irrelevant.

Bullshit. There is no contract involved if I fill an existing array with a series of random values.


I wanted this discussion to be civil, so I refrained from resort to restating your past history as >>113 did, but it's hard to do so when you keep confusing interfaces with implementations, and the "what should be" with "what is".

This has nothing to do with interfaces or implementations. This is abstracting shit you fucking idiot.

>>118
With that, shut up and go scrub another fucking toilet.