Also just about to apply for an apartment. Some stuff in the application I'll need clarification on and since it's past 5 my email won't likely get responded to until tomorrow but hopefully I'll be set to go then
Hmm... in order to prove whether an actual infinity can only be formulated in a circular fashion, I guess proving the contrapositive may be easier:
To make this independent of foundations, we need to first find a predicate in first order logic that defines (not necessary uniquely) an actual infinite object
I think P(x) where P = "is not finite" is sufficient. The challenge here is how to define "finite" in a foundationally independent manner under 1st order logic
In set theory alone there are already 5 notions of finite
What is observed so far is that things like natural numbers, dedekind finite sets, tarski finite sets and so on all shared some notion of limitation which make them different from infinity
If we can capture that limitation in first order logic than we may have a way to figure out whether defining actual infinity always require a self referential predicate
I am not sure what the formal name is, but what I mean is some predicate P where one of its parameters is x itself. Thus x is defined to be what satisfies P(x)
What I rithaniel and many others are suspecting is that infinity is something that can only be defined wrt itself, and cannot be defined via a "constructive" mean
But the answer to this question remains painstakingly open
unlike e.g. even numbers y are those that satisfies Q(y) where Q = "is divisible by 2", where here there is no circularity in Q since "divisible" and "2" can be formulated without reference to each other or the notion of "even"
anakhro: To clarify. We knew we can define an even number y by having y to satisfy "y is divisible by 2". Here there is no self reference and no underlying circularity because none of the concepts "divisible", "2" will reference the concept "even". That is, if you follow down the predicates A,B that defines "divisible", "2", you can avoid the concept "even" to appear in any such A,B
But the conjecture here is that, it is impossible for infinity to do so, meaning there is no way you can have a definition of infinity without it being referenced somewhere down the line
Because this question seemed to be still open. ZF and ZFC had not deal with this question directly because they have bypass it using the axiom of infinity (and large cardinal axioms when inaccessible cardinals were included), whereas Quine new foundations only push the thing higher into the universal set, which of course will contain infinity
Thus it will be interesting if the circularity of infinity can be rigorously proven independent of foundation systems under 1st order logic
Whether infinity can be defined using a predicate that is neither self referential nor has no underlying circularity in the sense illustrated in the "is divisible by 2" example
That is, whether infinity can be defined without invoking other notions of infinities, including the universal class
Yes they do. Because power set does not control the existence of all infinities (only the uncountable ones), it is axiom of infinity that we need to ask question about its nature
i mean, removing power set alone does not get us closer to the answer on whether infinity is non circular. Many constructivist still use some infinite sets such as the natural numbers and do not consider that circular
(also I am currently in seminar thus expect my response to delay)
In axiomatic set theory and the branches of mathematics and philosophy that use it, the axiom of infinity is one of the axioms of Zermelo–Fraenkel set theory. It guarantees the existence of at least one infinite set, namely a set containing the natural numbers. It was first published by Ernst Zermelo as part of his set theory in 1908.
== Formal statement ==
In the formal language of the Zermelo–Fraenkel axioms, the axiom reads:
∃
I
(
∅
∈
I
∧
∀...
That's the axiom of infinity.
"if there's really an infinity" <--- ??? what do you mean ???
One of my dreams is to reference the enormous real-world potential of the banach-tarski paradox in a grant application for research on the axiom of choice.
I'm looking forward to attaining an assistanceship
Though, I think the whole question about infinity is that the axiom of infinity is specifically there to say "let there be infinity." So the question is "Can you define infinity without axiomatically declaring it to exist and then prove that it does exist?"
I hold that you cannot, because using finite "tools" in a finite number of steps, you cannot attain an infinite object. There is a gap between the two and you can't cross it without making a jump.
The thing about the axiom of infinity that makes it unique is that all axioms which could potentially lead to the existence of infinity have to either directly or indirectly declare that infinity does exist.
Also, I agree entirely with your assessment, Anakhro. You do have to start somewhere.
(Though, I'm making pretty broad claims and I'm not sure I could back them up with rigor if it came to that.)
Perhaps I should say that it is a conjecture that the axiom of infinity has the property of not being derivable without an axiomatic declaration of the existence some infinite object or process.
1. No, I do not think there is anything wrong about axiom of infinity (I am an infinity lover, otherwise I would not spent so much time studying it)
2. As Rithaniel said, the unique thing about axiom of infinity, or any theorem or results that involves infinity seemed to always rely on that infinity or some universal object is already exist
3. The conjecture is then: can we find a system of axioms, none of which axiomise the existence of infinity, such that the existence of infinity can be derived from them
4. A much stronger conjecture, will be the additional requirement that the system of axioms has a bounded number of axioms
> Work in 𝖹𝖥−Infinity. If 𝜔 is a set then we're done, so assume that 𝜔 is a proper class (in which case it is the class of all ordinals.) Then there is a definable surjection 𝜔→𝑉.
Why we can assume $\omega$ exists as a proper class in ZF-infinity?
we did not proved the existence of $\omega$ first before using it?
anarkhro, rithaniel: Some partial answers to the conjecture (Rithaniel's which is my (3)): Dedekind infinite sets are those that self inject to a subset. The existence of such requires at least countable choice in ZF. I am not sure if there is a non circular way to derive the existence of a set that self inject to a subset because countable choice is just one of the many forms of "axiomatic declaration of the existence some infinite object or process"
@AjayMishra Such matrix is 2x3. I am guessing you already have 2 linear equations in 3 unknowns. Then the matrix is just the coefficients
Let T = (x,y,z) where x,y,z are vectors in $\Bbb{R}^2$. Then the two conditions means: x-z=(2,3) and 2x+y+3z=(-1,0). Thus solving them you end up still with a vector equation in 2 unknowns, meaning your possible T can span a plane
@Rithaniel The thing about the axiom of union that makes it unique is that all axioms which could potentially lead to the existence of unions have to either directly or indirectly declare that unions exist. Where's the difference?
I don't know what predicative means, I'm just talking about the ZFC axioms
Most of them assert the existence of some kind of set, which needs to be asserted by an axiom, in that regard I don't find infinity different from the rest
formal verification of FLT's proof is not a number theory goal, it is a goal for proof checkers, FLT is just a "long, difficult and new" proof in this context (with the added bonus of being super famous I guess)
Theorem in Rudin's FA: Let $X$ be an infinite dimensional Banach space and let $T$ be a compact operator. Then $0$ is an eigenvalue of $T$. Here's Rudin's proof: if $0$ is not an eigenvalue, then $R(T) = X$.
...?
I still haven't figured out why $R(T) = X$ is a contradiction, but I'm not worried about that at the moment. Why does $0$ not being an eigenvalue imply $T$ is surjective?
I can see that it implies that $T$ is injective...
Quote from Rudin's FA: "An operator $T \in B(X)$ is said to be invertible if there exists $S \in B(X)$ such that $ST=I=TS$. In this case we write $S = T^{-1}$. By the open mapping theorem, this happens if and only if $N(T) = \{0\}$ and $R(T) = X$."...So, $T$ is invertible if and only if $N(T)=\{0\}$ and $R(T) = X$? Why do we need the open mapping theorem to know this?
So, does anyone have spikes on the back of their head from using smartphones? I don't: I don't really use my phone; besides, my neck would be killing me if I had to look down at it for more than 5 minutes, so I would probably just find a way of supporting my neck if I had to use it, thereby preventing any spikes.
Yeah, that $N(T)=\{0\}$ and $R(T)=X$ are equivalent to $T$ being bijective is elementary, so that is where the subtlety in the definition of invertible lies.
@AlessandroCodenotti Yes, my statement was poorly phrased. It was me reading back on it that particular comment which spurred me to say that I was perhaps making claims which were too broad.
Though, it is worth noting that the axiom of union doesn't directly state the existence of unions in the context we usually think of them. It needs to be used in tandem with the axiom of pairing in order to result in "traditional" unions.
I was quite tired last night, too. Perhaps I should not have been engaging in discussion of axioms and logic.
I had a feeling all axioms except infinity can be derivable by axioms that does not reference any of the respective concepts (union, powersets, relations with bounded variables, empty set etc.)
Meanwhile, I think the easier subquestion that can help on that conjecture may be to check whether one can derive the folllowing predicate:
"A set that self injects into one of its subsets"
without using countable choice. I only knew that is impossible in ZF, but what is harder to examine is all possible set theories
Well, the tricky part is making sure your axioms are good axioms. Like, I could have an axiom that says "If you can define a list of elements, then there exists a set containing those elements" and then attain the concepts of union and intersection from that axiom, but it's sloppily worded (not that I can necessarily discern between sloppy wording and good wording, I'm just betting that it is sloppily written because I just now came up with it).
The axiom of infinity gives that you have an infinite set, not necessary for an object which is "infinite" in some philosophical sense. As far as I understand.
I still want to get a copy of the three volumes of the Principia Mathematica by Bertrand Russell to see what exactly they were doing that they took over 200 pages to prove 1+1=2
However, objects postulated in the other axioms seemed to be finitistic, so it seems there is a way to find an axiomatic system that can derive them non circularly (in the sense Rithaniel used when he state the conjecture)
> Work in 𝖹𝖥−Infinity. If 𝜔 is a set then we're done, so assume that 𝜔 is a proper class (in which case it is the class of all ordinals.) Then there is a definable surjection 𝜔→𝑉.
Would you consider an axiom saying "there is a function $f$ that maps dom(f) in dom(f) which is injective but not surjective" as directly referencing infinity?
anakhro: True, I was thinking a bit too far because when I am working on problems with my reverse mathematics hat on, I will not even declare something as a proper class if I cannot concretely construct it from something non circular, as otherwise it is easy to "cheat" by appealing to countable classes Alessandro: That's what I am thinking about when trying to tackle the subcase of dedekind infinite sets. The problem here is whether there is a way to prove that such f exists without referencing f or any notion of infinity. But yes that is a sufficient condition for referencing infinity
That I still need to think about it as I literally started thinking about this question today
hmm... sure, peano axioms seemed to be enough to assert the existence of a collection of natural numbers (thus $\omega$ can be defined using just a bounded predicate)
Using countable class still feels like cheating to me, I might need to think about this...
Hmm... so, we have the natural numbers constructed from peano axioms
and because of that, we can define the class $\omega$ using some definable formula
thus the class $\omega$ must exists and is not of the same size as any natural number (As otherwise, the successor of the current largest number will lead to a contradiction), which means it can only be infinite (countable to be specific)
Hmm interesting, so a countable class is definable in 1st order logic
While now convinced that a countable class is definable without circularity, I do think using some axioms that derive "there is a function f that maps dom(f) in dom(f) which is injective but not surjective" seemed to be even better (because then not even countable is used), but that might be too harsh and I probably need to spend some time thinking about how to do that
Short summary of today's investigation on the nature of infinity:
It is not size nor invariance nor unboundness that characterise what is meant by something to be infinity. It is some notion of incompleteness and unreachability (that you cannot get the whole thing starting with any parts of it). Will investigate this more as I continue on the RudyRucker book as well investigating on the question on how to construct the aforementioned function f
Could someone provide a reference that includes a full and honest proof of the Correspondence Theorem for rings?
Let $A$ be a multiplicative ring with identity and $I$ an ideal of $A$. There is a one-to-one correspondence between the ideals of $A$ that contain $I$ and the ideals of the quotie...
In fluid dynamics, a stagnation point is a point in a flow field where the local velocity of the fluid is zero. Stagnation points exist at the surface of objects in the flow field, where the fluid is brought to rest by the object. The Bernoulli equation shows that the static pressure is highest when the velocity is zero and hence static pressure is at its maximum value at stagnation points. This static pressure is called the stagnation pressure.The Bernoulli equation applicable to incompressible flow shows that the stagnation pressure is equal to the dynamic pressure plus static pressure. Total...
Hey, is there a standard notation for specifying particular roots of particular polynomials?
I see that, for example, $x^6 - x - 1$ has a root that is approximately $1.13472$, but is there a more concise and standardized expression for this number than "the root of $x^6 - x - 1$ that is approximately $1.13472$"?
@TannerSwett I doubt there's a systematic notation, given the sheer breadth of possibilities. But in your case I think the simplest one would be "the unique positive real root"
good evening. Im a bit confused. Will a converging series of cosines, which involve non-integer spaced frequencies always yield a function that comes arbitrarily close back to its initial condition?
i guess this might be true since if it converges, at some point the next oscillations don't matter much anymore. So it basically reduces to a finite sum of oscillations, which will "rephase" like this
I'd say yes. There's guaranteed to be a point where the 10 largest terms all coincide to within 0.1 degrees, as well as a point where the 100 largest terms all coincide to within 0.01 degrees, and so forth.
yes. as long as there is a finite sum of oscillations, they will come arbitrarily close if you wait long enough. but what if you add infinitlly many? strange things can happen then right?
Right, but you can simply identify how close you want to come (say, you want to find a place that mirrors the initial condition to within 0.01%), and then you can just take enough terms that you've captured 99.99% of it.
i guess you can choose an N large enough, so that the contribution of all terms after that become smaller then a certain value for all times. then the oscillations will coincide to a range of degrees, arbitrarily close to that vlaue
You'll get a certain amount of error due to truncating the series, as well as a certain amount of error due to the phase not exactly matching, but both of those can be made arbitrarily small, at the same time.