Empty Set doesn't exist

If you cant sense it, then it doesnt exist.

You cant see emptiness, therefore emptiness doesnt exist.

>>38
Using a library? Guess what?
0/10

http://common-lisp.net/project/cl-containers/

>>41
No, I did not use a library.  I created a datatype without any constructors.  Something seemingly pointless, like empty sets.

>>42
http://hackage.haskell.org/packages/archive/Cardinality/0.2/doc/html/Data-EmptySet.html

>>43
That's a set with a single element.

>>44
Nope.

How are you sure other people exist?

>>46
"I think, therefore I am" is incorrect because no one on here thinks.

>>38
That's not empty, it still contains bottom.
notTheEmptySet :: EmptySet -> EmptySet
notTheEmptySet = notTheEmptySet

And, >>37 is right, '() corresponds to the empty set.
We can define a linked list as a series of conses where:
F(()) -> {}
F((1 . x)) -> {{1}, F(x)}

>>48
That's like saying "qzmogsdfjzeogwns" corresponds to the number 16 because of its length. The list-to-set conversion you're using is completely arbitrary and probably isn't the only possible one.

>>37
No, you are shit at comparing things. Here's Python's interactive environment doing the same thing:

Python 2.7.2 (default, Jun 12 2011, 15:08:59) [MSC v.1500 32 bit (Intel)] on win32
Type "copyright", "credits" or "license()" for more information.
>>> set()
set([])
>>>

>>46
Somebody have to make all these posts.

>>42
Something seemingly pointless, like empty sets.
Haskell: where everything seemingly pointless, like empty sets.

>>52
2deep4u

>>53
[cpde]
$ ghci
GHCi, version 6.8.2: http://www.haskell.org/ghc/  :? for help
Loading package base ... linking ... done.
Prelude> id id id id id id id id id id id id id id id id id id id id id id id id id 0
Fuuuuuuuuuuuuu...
Interrupted.
[/code]

>>54

Lisp                              | Haskell
----------------------------------|---------------------------------------------
(defun id (x) x)                  | Prelude> id id id id id id id id id id id id
(id (id (id (id (id (id (id (id   |   id id id id id id id id id id id id id 0
 (id (id (id (id (id (id (id (id  | <interactive>: out of memory
  (id (id (id (id (id (id (id (id |
   (id 0))))))))))))))))))))))))) |
0                                 |


C
------------------------------------------------------
id(id(id(id(id(id(id(id(id(id(id(id(id(x)))))))))))));
Segmentation fault

Do not argue with the ultrafinist, or WWBT! (I should know better, for I'm about to)

He doesn't even believe in the concept of natural numbers, much less in any more complex abstract concepts (such as set theories).
The only thing he seems to believe in are his direct senses, but he refuses to actually infer (by induction or deduction) anything from them.
He claims mathematics is religion, and I'll partially agree with him on this(more on this soon), although mathematics just teaches "If A then B" and it doesn't force you to believe in some particular A. Mathematics proves very useful in the real world and thus the consequences and things it teaches have a good chance of being true. There is another way to look at this: claim that computation exists (in the Turing-equivalent sense), and you also get most mathematical theorems from that. The thing about computation is that it just means that simple finite abstract rules can be followed and will always give the same result in all possible worlds. It can be shown that computation can be encoded in arithmetic and a Turing Machine can also do arithmetic (unbounded). The Chuch-Turing Thesis shows that computation is universal and could even be considered a natural class. If you posit a reality having well-defined physical laws, you'll also have to posit some form of computation (our world is capable of it, at least in the Finite State Machine sense, a restricted Turing Machine). If after some thought experiments you end up subscribing to functionalism or computationalism, you also have to put computation in the ontology (actually it's the only thing that you truly require). Such beliefs, be they in a reality, your own mind or in the fact that computation works, or in mathematical truths are "religious" beliefs, although they do have substatial evidence for them being true - in another way, some parts of math are good things to bet on being true. If you ask me a belief in the Chuch Turing Thesis is a good "religious" belief - a bet on something being true(even if supported by evidence), such a bet taken more seriously can lead you to some forms of Platonism (again, religious belief). Some beliefs are incompatible, for example, the belief in functionalism and a negative belief in computation, or a belief in an ontologically primitive reality and computationalism (the version where one assumes one's mind exists), and one should not hold incompatible beliefs if they want to be correct.

Me, I have no problem in making such bets (about certain things being true) when I have good evidence for them. Why? I risk being more wrong and stand to gain getting closer to more true statements. Of course, a smarter way to go about this is to do uncertain inference: hold confidence values for each belief and adjust them based on evidence.

Too bad our local ultrafinitist doesn't want to have any beliefs, and so he will never ever have a chance of being right or wrong. He may hold various beliefs, but will refuse to acknowledge that he holds them. In other cases, he must bet on some theory, and if he bets or not is itself a choice, and not betting is also a form of "belief" (or lack of).

Good luck to you, ``in LISP''/symta-guy, and I hope you do some more thinking about what beliefs are acceptable to you and which are not, as well as what standards you use to select them. It's easy to pick some ultraskeptical standard which doesn't allow you to hold any beliefs at all, but that limits greatly one's abilities, to the point of never being able to be right or wrong, or making any choices. I think it's great having a chance of being right, and if you're wrong, that's not a problem as long as you retain the ability of recognizing wrong beliefs and removing/replacing them, thus being less wrong as time passes (and now I expect that you'll dispute time's existence - that can be done as well, but again limits your options!).

>>57
The only thing he seems to believe in are his direct senses, but he refuses to actually infer (by induction or deduction) anything from them.
No. I'm don't refuse. I just want the inference to be justified by senses too. The inductive proposition that "given N, there exists N+1" cant be justified.

>>58
I just want the inference to be justified by senses too.
The problem with this is that you won't be able to form any theories/metatheories about why you have senses at all, or why physics (or just whatever makes your senses be this or that) behaves the way it does and so on.

That inductive proposition talks about abstract systems, in the real-world, you will have a local limit imposed by physics.

If for example you deduced by induction that reality corresponds to some specific mathematical structure. Why only that structure? Let's say its information limit is some constant k. Why not k-1 or k+1? Occam's Razor suggests that the simplest theory is that all possibilities are realizied, thus also the k+1 world. A belief in only world k is a stronger belief than a belief in all finite worlds 0,1,...,k,k+1,...
It's a stronger belief just like the belief that the sun doesn't exist as long as you don't look at it(such as at night).

Math itself tries to be as general as possible - in that it can also be used by physics, but general enough to not limit itself to only what's locally physically possible. There is no reason why any finite natural number should be the limit, hence why you get a countable infinity of natural numbers.

You could say that all this is abstract non-sense, and that you don't care about why anything behaves like it does.
Consider this then: in the far future, you're very old and have developed some incurable brain tumor, the doctor offers you to get a digital brain replacement (be it gradual or instant, this is a thought experiment, so pick whatever suits you best), do you say yes or no? You have to bet on a theory because your future experiences depend on it, yet without having worked out all the possibilities you cannot make a choice, but you don't have the luxury of not making a choice - a default no also has costs.

>>59
Why are you arguing about math with someone who has no expectation that the sun will rise yet again tomorrow? (I could ask him the corresponding question, but it would be less fruitful.)

>>59
The problem with this is that you won't be able to form any theories/metatheories about why you have senses at all, or why physics (or just whatever makes your senses be this or that) behaves the way it does and so on.
Why should we form useless theories, which are completely unrelated to reality?

That inductive proposition talks about abstract systems, in the real-world, you will have a local limit imposed by physics.
Why do we need "abstract systems" or any other religion?


If for example you deduced by induction that reality corresponds to some specific mathematical structure.
I've a better idea! Lets find correspondency to some physicaly sensible structure, and throw the mathematics aways.

Why only that structure? Let's say its information limit is some constant k. Why not k-1 or k+1? Occam's Razor suggests that the simplest theory is that all possibilities are realizied, thus also the k+1 world. A belief in only world k is a stronger belief than a belief in all finite worlds 0,1,...,k,k+1,...
Does Occam's Razor suggests it's own validity. and how do define the "simples theory". For a religious man/woman the term "God" maybe simple (it's much "simpler" than your phyisical theories, Einstein).

It's a stronger belief just like the belief that the sun doesn't exist as long as you don't look at it(such as at night).
It exists in my memory (or machine's memoty, if we talk about AI).

Math itself tries to be as general as possible
You found the main problem with math. There is no such thing as "general" or "all" or "everything", without some kind of bounded universe, like cons-pair list, one passes to Lisp's `all` and "any" high-order function.

in that it can also be used by physics
Can you imagine a “physical process” whose outcome could depend on whether there’s a set larger than the set of integers but smaller than the set of real numbers? If so, what would it look like? -- Scott Aaronson

There is no reason why any finite natural number should be the limit, hence why you get a countable infinity of natural numbers.
There is no reason why physical world should be the limit, hence why you get a heaven with almighty God.


You could say that all this is abstract non-sense
Mathematicians themself call it nonsense:
http://en.wikipedia.org/wiki/Abstract_nonsense
There is good reason for that - it cant be justified by senses.

Consider this then: in the far future, you're very old and have developed some incurable brain tumor, the doctor offers you to get a digital brain replacement (be it gradual or instant, this is a thought experiment, so pick whatever suits you best), do you say yes or no? You have to bet on a theory because your future experiences depend on it, yet without having worked out all the possibilities you cannot make a choice, but you don't have the luxury of not making a choice - a default no also has costs.
How that is related to the question of existence? In both, "yes" and "no" cases, it wont contradict my point of view. Doctor just makes a replacement copy, like you copy audio CDs. No magic or paradoxes.

Python, Lisp, and Haskell are shit.

>>60
Why are you arguing about math with someone who has no expectation that the sun will rise yet again tomorrow? (I could ask him the corresponding question, but it would be less fruitful.)
Do you have strong evidence, that sun wont rise? Why not? I think you're lying, mathematician!

>>62

Lisp   | Haskell
-------|------------------------------------------------------------------------
lambda | inference, lambda cube, strongly normalizing, equality-qualified types,
       | algebraic types, existential types, phantom types, dependent types,
       | higher-kinded types, linear types, affine types, unique types,
       | nominal types, signatures types, recursive types, type classes,
       | type annotations, principal types, higher-order abstract syntax,
       | generalized algebraic types, robinson's unification, hindley-milner,
       | constrained types, polymorphic recursion, parametric polymorphism,
       | equivalence classes, type order, judgments, curry-howard isomorphism,
       | system t, system f, products, coproducts, categorial sum, call-by-name,
       | inhabited types, higher-rank impredicative polymorphism, covariance,
       | subtype polymorphism, ad-hoc polymorphism, predicative types,
       | bounded quantification, contravariance, inductive types...

>>64

C      | Haskell
-------|------------------------------------------------------------------------
void * | inference, lambda cube, strongly normalizing, equality-qualified types,
       | algebraic types, existential types, phantom types, dependent types,
       | higher-kinded types, linear types, affine types, unique types,
       | nominal types, signatures types, recursive types, type classes,
       | type annotations, principal types, higher-order abstract syntax,
       | generalized algebraic types, robinson's unification, hindley-milner,
       | constrained types, polymorphic recursion, parametric polymorphism,
       | equivalence classes, type order, judgments, curry-howard isomorphism,
       | system t, system f, products, coproducts, categorial sum, call-by-name,
       | inhabited types, higher-rank impredicative polymorphism, covariance,
       | subtype polymorphism, ad-hoc polymorphism, predicative types,
       | bounded quantification, contravariance, inductive types...

>>58

No. I'm don't refuse. I just want the inference to be justified by senses too. The inductive proposition that "given N, there exists N+1" cant be justified.

You've never seen dominoes fall?

>>61,64-65
Now I remember why I hate the ``in Lisp'' guy.

>>57,59
I like you. Don't let him troll you, it hurts to see.

>>66
There are finite number of dominoes.

>>61

Why should we form useless theories, which are completely unrelated to reality?

If you would just fucking study it, you would learn about the discoveries that have been made, and the applications that were found form them years or decades after their discovery. You are using your ignorance as justification for its lack of worth.

>>68

not in my playroom. I have infinite dominoes, and I watch them topple all day long, for eternity.

>>61
I've a better idea! Lets find correspondency to some physicaly sensible structure, and throw the mathematics aways.
I have no idea how you could talk about it without some form of math.
Does Occam's Razor suggests it's own validity. and how do define the "simples theory". For a religious man/woman the term "God" maybe simple (it's much "simpler" than your phyisical theories, Einstein).
You could define it by algorithmic complexity, such as Kolmogorov complexity. God is very complex in popular definitions and almost everyone has different definitions.
There is no reason why physical world should be the limit, hence why you get a heaven with almighty God.
There are many more choices than that. There are also concepts which are logically inconsistent thus cannot have any form of existence (except as an inconsistent idea). Consider for example the position that computation is the only thing contained in the ontology. If that is the case, there are ways one could show that other worlds which are non-physical (in the local sense) can exist and can be accessible, if you want an example of how this could be practically realized, you should read Greg Egan's "Permutation City". However, if you privilege only some particular structure with "existence" (typically called physical nature), then that would no longer work, but you'll have a more complex theory that privileges only this single world with existence. Where do you stop? What interpretation of quantum mechanics do you choose? Not all choices are compatible with each other. Each assumption has very specific consequences.

How that is related to the question of existence? In both, "yes" and "no" cases, it wont contradict my point of view. Doctor just makes a replacement copy, like you copy audio CDs. No magic or paradoxes.

Replacement copy? Would 'you' still be conscious? Would you have some form of contintuity of consciousness? Would it change the nature of your senses? If you assume some multiverses (which are actually forced upon on you, if you assume both computationalism and that you have a mind, but since you refused to reason about it, you're not aware of such consequences), do you predict that your future experiences to be in the same universe and that they would be stable (what about changes in measure)?

These things are not trivial. If you assume you have conscious experience/senses, that has consequences. If you assume it's an illusion, then there are different consequences.

If your belief is non-computationalist, you would believe that if you say 'no', you still have a survival chance, but saying 'yes' kills you. If you have a computationalist belief, you can say 'yes', but you can never know for sure that your beliefs are correct (despite appearing internally consistent), thus in either case it's a "religious" bet. Or was your question about what does this have to do with the existence of natural numbers? A belief in computationalism imposes very strict requirements about what one's ontology has to be (due to the Chuch Turing Thesis and what is known as the "Universal Dovetailer Argument" and "Movie Graph Argument(1-3)"). I could elaborate on this, but it would be less effort for both you and me if you just used google and looked up the arguments by yourself as they are way off-topic to this thread.

>>67
I should stop responding, but at times it makes me think that he probably believes what he's saying, which makes me wonder if he realizes that his beliefs might not be consistent(free of contradictions).

>>1 but you can sense space? space can be empty, but there is still space... (ever noticed how the walls don't cave in, in an empty room? ^^)

so how many empty spaces does an empty set have?

btw wouldn't the definition of set be Two or more items? A set of one isn't much of a set, a set of none even less so...

I'll show you abstract nonsense ^^

Can you have negative one item(s) in a set? (Eg. maybe something that isn't in the set?)

would the empty set have a minimum bound negative infinity / maximum bound 0? (contains everything that ISN'T in the empty set??)

>>72
There is no light without dark. You can sense emptiness only relative to something.

>>70
not in my playroom. I have infinite dominoes, and I watch them topple all day long, for eternity.
You have autism. Also, note, in some countries, the term "autism" is considered to be a lighter form of schizophrenia, because it sometimes progresses into it. So you are a potential schizo/psycho.

If you would just fucking study it
How can one study something, he doesn't understand. "Set Theory" is axiom, meaning one should have inborn understanding of it. I dont have. I cant imagine "a set", this highly abstract, invisible, unordered and unbounded entity.


I have no idea how you could talk about it without some form of math.
Lisp's lambda (in it's call-by-name form) allows proving sensible theorems without any math. And lambda is a physical entity. Of course lambda has its "problems", but these problems manifest themselves only in the presence of "infinity", meaning they aren't really problems, but the way systems shows us, that we doing something wrong.

You could define it by algorithmic complexity, such as Kolmogorov complexity. God is very complex in popular definitions and almost everyone has different definitions.
Sorry, I dont understand terms "algorithmic complexity" and "Kolmogorov". The notion of "mathematical complexity" is by itself pretty complex for my little goy brains.

There are many more choices than that.
For me, there are only choices I see. Making hypotheses without confirming data would be a waste of time. Even the most crazy projects, like SETI, have some confirming data.

Replacement copy? Would 'you' still be conscious?
What is "consciousness"? It's a vague term. I'm afraid, you yourself don't see what you're talking about.

I should stop responding, but at times it makes me think that he probably believes what he's saying
I do believe what I'm saying, because I havent seen better theory, than subjective idealism.

which makes me wonder if he realizes that his beliefs might not be consistent(free of contradictions).
"consistency" is a buzzword and in reality, there are no contradictions, except those we invent/define.

>>74

Well if you don't like math, then feel free to not study it. Life it too short to base it around things you'd rather not do. But you should acknowledge that there are things about math that you are not yet aware of, both in itself and in its nature. There is nothing concrete about math. It appears very formal, but it was devised by people, and over a very long period of time, was unified and reinforced with a standard vocabulary. Of course, you need to stay very formal or else you will quickly get lost, and you wont be able to get your bearings. Logic forms the ground at your feet. If you get lost, you can look at the ground and see where you can immediately walk. But if you want to get somewhere you will have to see the big picture. But if you can't see the ground you are walking on, you'll probably trip and fall into a cave of incorrectness. Then you need to slowly retrace your steps until you arrive at where you fell, and then ponder what the right step should have been.

>>74
For me, there are only choices I see. Making hypotheses without confirming data would be a waste of time. Even the most crazy projects, like SETI, have some confirming data.
Actually there are physical evidences/confirming data for some such theories, however science can never "prove" anything, it can only disprove some false theories.
What is "consciousness"? It's a vague term. I'm afraid, you yourself don't see what you're talking about.
The fact that I have unified experience: sound, images, other senses; that such senses have some continuity over (subjective) time. If you think the term is nonsense, just say so. There is just one problem with this: you infer the material world's existence through your senses (by induction) and then from that inference you see that anyone claiming to be conscious is delusional about it as there's no such thing as mental experience in the material world, except you just used your own senses to infer the material world's existence, so it's self-defeating. There are also more subtle contradictions with this as shown by the "Movie Graph Argument" or "China Brain" thought experiment (the only problem is that in the China Brain example, the author does not see that there is a solution to the problem which is free from absurdities).
I do believe what I'm saying, because I havent seen better theory, than subjective idealism.
My favorite candidate for now is a form of "arithmetical"/computationalist neutral monism (in it, both mind and matter are a consequence of "immaterial" natural number relations). Unlike a lot of philosophical/metaphysical theories, it makes some very solid predictions about the shape of physical law (confirming some of the laws of quantum mechanics) as well as future expected experiences (some of which let you set it apart from other theories, thus making it falsifiable). Is it possible that I am wrong? sure, no beliefs or mine are absolute, it's just the one which I have the highest confidence value for now, if it's wrong, then some extra "magic" will have to exist to explain both physical and mental phenomena, and some rather counterintuitive things will be possible (beyond what most scientists expect today). Why this theory and not another? It follows as a consequence of a few rather simple assumptions, but any assumption has a chance of being wrong, no matter how much it makes sense to me right now. Still, I must bet on what makes most sense to me, and I don't have any problem with that.

Anyways, the problem with subjective idealism is that it's not really falsifiable (unlike my form of neutral monism) and it cannot be used to predict anything. What's worse is that it's a solipsistic theory - it only says that you have mind, it doesn't claim that I have mind. In my case, I can predict that both of us have minds and even a wide array of properties of those minds, as well as what the physical world is, why it exists, what is its relation to one's mind, why does that exist and so on.

"consistency" is a buzzword and in reality, there are no contradictions, except those we invent/define.
I didn't say that reality is inconsistent. I think that whatever reality is, it cannot be inconsistent. It's not a buzzword, it has a very specific meaning: a contradiction or paradox in beliefs - it happens when you prove that something is both true and false starting from some assumption or axiom. In the real world, it could be something like someone holding contradictory beliefs, such as "this person is my father and is not my father", where each of the terms in that sentence have un-ambigous meaning within that person's mind.
except those we invent/define.
Of course contradictions are in our own mind - they just show that some of our reasoning or beliefs are wrong and we must re-examine them.

Hmm..

Set A +{1,2,3,4} / -{5}

Subset B of A +{1,2} / -{3, 4}

Not (Not B) == B (tadaa ^^)

...while i'm at it

Set A is Not a Subset of Set A

Set A IS SET A

"Set Theory" is axiom, meaning one should have inborn understanding of it. I dont have. I cant imagine "a set", this highly abstract, invisible, unordered and unbounded entity.
If you can't while others can, then you are defective.

>>76
Of course contradictions are in our own mind - they just show that some of our reasoning or beliefs are wrong and we must re-examine them.
But mathematicians love to "define" contradictions. When I hypothesise that I can move hand into some occupied space, and then fail to do this - that is an invented contradiction. When mathematician proposes that moving hand in some (un)occupied space is a contradiction (by sole definition), that would be a defined contradiction, that has no sense, because it's in strict contrast with senses.