/sci/ - Science & Math

>>10817722
>but can you enumerate every conceivable formal system?
Yes. Proof: dude, just think about it.

>>10817722
>we know that you can, through godel numbering, enumerate every wff in any given formal system, meaning they are actually countably infinite, but can you enumerate every conceivable formal system?
not science or math

>>10817985
It's math. It's metamath.

>>10817722
Yes, I can. What's the pay for being an N-Set descriptor practitioner?

>PRO=TIP: If it is less than 7 figures then Godel was a fucking hack mathematician that just cucked everyone else into believing his bullshit

>>10818024
I think you're in deep waters with this one, Solivagus.

>>10818038
I am the deep that calls, for I am buried far lower than you could ever measure.

I am the memory that echos back, I am the God of Pleasure.

https://www.youtube.com/watch?v=MLAORGaM-g4

>>10818043
And he's back!

>>10818046
Ready for your Anonymous heart-attack?

>Let's a GO! IT'S A ME! MARRY-A-HO!

>>10818068
Asshole

>>10818088
What is a digital proctologist doing teaching their University class here on 4chan?

>boohoo government/mods/police won't save any of you from me!

>>10817813
seriously, how?

>>10818163
Make me a fucking tangible offer with a good faith payment of $40,000,000 USD up front.

>I have broken BTC encryption, Harmony ONE is the only one that does not work off the same principle. It is the only safe coin in existence.

Be more precise.
Do you mean enumerate them all at once (in one go)? What do you mean by enumerating a formal system? It's axioms? If the axioms are schemas ("for propositions p, it's an axiom that A(p)"), how do you treat this?

>>10817722
Each formal system uses primitive symbols (which collectively form an alphabet) to finitely construct a formal language from a set of axioms through inferential rules of formation.

The system thus consists of valid formulas built up through finite combinations of the primitive symbols—combinations that are formed from the axioms in accordance with the stated rules.[3]

More formally, this can be expressed as the following:

A finite set of symbols, known as the alphabet, which concatenate formulas, so that a formula is just a finite string of symbols taken from the alphabet.
A grammar consisting of rules to form formulas from simpler formulas. A formula is said to be well-formed if it can be formed using the rules of the formal grammar. It is often required that there be a decision procedure for deciding whether a formula is well-formed.
A set of axioms, or axiom schemata, consisting of well-formed formulas.
A set of inference rules. A well-formed formula that can be inferred from the axioms is known as a theorem of the formal system.

>>10818185
https://www.youtube.com/watch?v=BELlZKpi1Zs

>>10818191
>X is for box

>>10818191
you didn't answer my question

https://youtu.be/Q53GmMCqmAM

>>10818185
>A finite set of symbols, known as the alphabet, which concatenate formulas, so that a formula is just a finite string of symbols taken from the alphabet.
>A grammar consisting of rules to form formulas from simpler formulas. A formula is said to be well-formed if it can be formed using the rules of the formal grammar. It is often required that there be a decision procedure for deciding whether a formula is well-formed.
>A set of axioms, or axiom schemata, consisting of well-formed formulas.
>A set of inference rules. A well-formed formula that can be inferred from the axioms is known as a theorem of the formal system.
so can i hit all these things (ie every finite possible discrete number) for every finite combination of symbols, rules, grammars and axioms?

>>10818185
consider some formal system F1

5 symbols, 5 grammar rules, 5 axioms, and 5 inference rules. can i numerate every single formal system conceivable by just iterating through this?

>>10818279
fix length n = 1
run through all 5^1 possible letters
up n to 2
run through all 5^2 possible letter tupples
up n to 3
run through all 5^3 possible letter tripples

You run through e.g. a tripple at an alphabet of three leters (making for 3^3 possibilities), by varying the last element quickest, the second to last one second quickers, and so on, e.g.


(a,a,a)
(a,a,b)
(a,a,c)
(a,b,a)
(a,b,b)
(a,b,c)
(a,c,a)
(a,c,b)
(a,c,c)
(b,a,a)
(b,a,b)
(b,a,c)
(b,b,a)
(b,b,b)
(b,b,c)
(b,c,a)
(b,c,b)
(b,c,c)
(c,a,a)
(c,a,b)
(c,a,c)
(c,b,a)
(c,b,b)
(c,b,c)
(c,c,a)
(c,c,b)
(c,c,c)

And for each you run a check if it's a wellformed word (e.g. "abcab" might be valid for your system) and discard it if it isn't.
Btw. you can get away with only one rule, e.g modus ponage.

>>10818302
>modus ponage
top zozzle

>>10818302
Here's a short python routine to get those

def words(smybols, length):
ws = [[]] # initialize
for _ in range(length): # consider length iterations
ws = [w + [c] for w in ws for c in smybols] # run through existing w's and add symbol
return ws

for word in words('abc', 4):
print(''.join(word))

(I will work this into a short related video later)

>>10818302
caveat emptor > tempus ponens > modus pownage < Solivagus's harem-slut scientist girls.

Boom, Sindy and all of mine are now the intellect running this simulation/emulation.

>KING ME BITCHES!

Ya fucking n00bfag.

Yup. All of science is my personal responsibility. Why? Because none of you were willing to breath and die with every moment, memory, and master.

>Martial Arts LCD STICKING PLASTER!

>>10817985
t. Doesnt actually study or have a degree in science or math

>>10818279
>>10819355
there you go:

https://youtu.be/0HtZclfnnQw

>>10820027
oh dang, thank you
i like the channel too

>>10820027
from a purley set theoretic conception, is this possible with an infinite alphabet?

>>10819909
I think they’re suggesting mathematical logic and analytic philosophy are not science or math.

>>10820107
thx, I plan on doing more set theory eventually

>>10820110
[math] \omega^\omega [/math] (where you may think of [math] \omega [/math] as the naturals) is much less worse than one might think. In fact consider
https://en.wikipedia.org/wiki/Epsilon_numbers_(mathematics)
But don't confuse with the corresponding cardinal expressions.
See also
https://en.wikipedia.org/wiki/Ordinal_arithmetic#Exponentiation

>>10820193
emphasis on even ω^ω^ω^... being countable.

>>10820159
Yeah its just a weird statement, because (1) recursion theory is primarily studied by computer scientists, and some mathematicians, (2) these techniques come up in areas besides recursion theory or logic, for instance topology often uses fixed points and diagonalization, and pretty much any logic can naturally be interpretted as a model of an algebra, so there's currently a ton of research going on relating to different models to different logics, and a single logic can have many distinct interpretations.

This is especially true of modal logics, but other logics too. E.g. classical logic corresponds to Lindenbaum-Tarksi algebras, intuitionistic logic to heyting algebras, "normal" modal logic to interior algebras. Furthermore, category theorists and people working on coalgebra are interested in modal logics because they provide a very natural and concrete perspective on something called bisimulation (which I know nothing about).
Then so-called "non-normal" modal logics have tons of interplay going on with all sorts of computer science, cognitive science, and applied math. For instance the 'box' operator can easily be used to capture most of the properties of what are called closure spaces and subset spaces.

>>10820199
>>10820193
im a literal brainlet here, how would i list every possible string taken from an infinite alphabet? It just seems that you could diagonalize and find some other string. its just not clicking for meh
thanks alot for the help btw

>>10820686
I'm not willing to go too into it, but there does exist an injective map of the all the strings of an alphabet A onto the natural numbers (we will call this A*). This also means that A* is at most countable.

>>10817722
no, you can trivially add any subset of the natural numbers as constants to Peano arithmetic, and there are uncountably many subsets of natural numbers

>>10820596
Don’t waste your breath trying to explain to these idiots/trolls why mathematical logic is math.

>>10820709
>you can trivially add any subset of the natural numbers as constants to Peano arithmetic
You don’t know what you’re talking about. What you just said is nonsense. Alphabets are finite.

>>10817722
wasnt there a better thread about this a few weels agp?
>>/sci/thread/S10742531

>>10820853
but none of that is right
the graph method dosent hit every formal system at all because you can code the nodes and jlist them to diagonilize

and the other retard is just talking about logic

>>10820686
>diagonalize
The diagonalization construction would give you a tuple of the length of the site domain of the diagonalization function.
That's why I said don't confuse it with the cardinal expression of exponentiation.
Ordinal w^w doesn't require you to go from w to w. It doesn't hold a tuple with infinite length.
Instead you have tuples of always growing length and always growing quantity of used symbols.

Look at the linked Exponentiation section of the ordinal arithmetic Wikipedia article.
There's even ordinals beyond w^w^w^w^... which are still countable.
It's not about objects that look like real numbers/infinite floats, instead it's about infinitely deep nestings of counting tuples of tuples.

Related (but not to your question)
https://en.m.wikipedia.org/wiki/Gentzen%27s_consistency_proof

of the size of the domain*

>>10820686 #
>diagonalize
The diagonalization construction would give you a tuple of the length of the size domain of the diagonalization function (I.e. length of |w|)
That's why I said don't confuse it with the cardinal expression of exponentiation.

Ordinal w^w doesn't require you to go from w to w. It doesn't hold a tuple with infinite length.
Instead you have tuples of always growing length and always growing quantity of used symbols.

Look at the linked Exponentiation section of the ordinal arithmetic Wikipedia article:
https://en.wikipedia.org/wiki/Ordinal_arithmetic#Exponentiation
There's even ordinals beyond w^w^w^w^... which are still countable.
Because it's not about objects that look like real numbers/infinite floats, instead it's about infinitely deep nestings of counting tuples of tuples.

Related (but not to your question)
https://en.m.wikipedia.org/wiki/Gentzen%27s_consistency_proof

But in relation to the original question, I should point out that there is in principle conceptualised also
https://en.m.wikipedia.org/wiki/Infinitary_logic

Shitposting language poison into /sci/ graduates really is an effective tactic. It's like preying on abandoned orphans of intellect, yumyum

good thread

>>10820847
fine, then pick some subset of the natural numbers and call those the theorems of a formal system, and let the alphabet be 0's and 1's and interpret each string as a natural number
again, there are uncountably many subsets of the natural numbers, therefore uncountably many formal systems
this is an injective map from the power set of the set of natural numbers to a set of formal systems

Do you think a countable union of countable sets should be countable?

Or, more violently, would you without question adopt the axiom of countable choice (that implies the former)?