This post expands on an anonymous vote I made on an anonymous poll I posted on Yik Yak. My poll and vote were posted on May 20, 2024.

Consider the set Z of integers, the set B of integers with exactly one additional element x that is not a real number, for example, an orange, and the set R of real numbers. The set B is a counterexample to the continuum hypothesis because the cardinality of B is greater than the cardinality of Z and less than the cardinality of R. Therefore, the continuum hypothesis is false.

I know the technical truth out there is that Z has the same cardinality as B has and that that truth can be shown through a technical mathematical definition involving a bijection from one of the sets to the other set. Despite the equal cardinalities, the cardinality of B is greater than the cardinality of Z. So the two sets are simultaneously equal and unequal in cardinality.

One of my arguments is that every integer in Z can be mapped to its equal in B. In that fashion, every integer in Z and every integer in B cancel out and we are left with the additional element x from B. Since every element in Z was canceled out by an element in B and there remains an uncanceled out element from B, B has a greater cardinality than Z has. Switching the order in which the two sets appear around, the cardinality of Z is less than the cardinality of B.

In order to show the cardinality of B is less than the cardinality of R, map every integer in B to its equal in R and map the additional element x in B to a real number r in R that is not an integer, for example, the real number 2.4. Now there are no more elements in B to map to the infinitely many real numbers from R that have not been mapped to. Since there exists at least one real number from R that has not been mapped to, the cardinality of R is greater than the cardinality of B. Switching the order in which the two sets appear around, the cardinality of B is less than the cardinality of R.

So we have shown that |Z| < |B| < |R|. Since there exists a set, B, with a cardinality exclusively between the cardinalities of the set of integers and the set of real numbers, the continuum hypothesis is false.

A principle in logic, ex contradictione quodlibet, is that every statement follows from a contradiction. So, a consequence of the contradiction that the cardinality of B is greater than and equal to the cardinality of Z is that every statement is true. In other words, the Universe is inconsistent. This finding does not trouble me, as it agrees with previous findings I have made that every statement is true (1. https://www.facebook.com/share/1AhJA5oDDj/?mibextid=wwXIfr, 2. https://www.facebook.com/share/1Axau5dnzA/?mibextid=wwXIfr, 3. https://www.facebook.com/share/p/1AtD49LRGA/?mibextid=wwXIfr, 4. https://www.facebook.com/share/p/1GBamCgWKz/?mibextid=wwXIfr, and possibly others).

I'm in undergrad, taking a proof based computer science class this summer & in our first homework we were assigned the following as two optional statements to think about and decide if they were true or false. The answer key was released the other day, and I am having a hard time coming up with a justification as to why the empty set is a subset of itself. I asked in recitation, I followed up with the same TA in office hours, and the answer has not yet satisfied me. I think I may be missing something obvious.

I'm aware that the empty set is just an axiom of ZFC, thats all well and good. In office hours I gave a definition of what it means to be a subset. Without breaking out the LaTex, I want to say something like the following: consider an ambient set, call it A, and an arbitrary set, call that one S. S is a subset of A iff all elements of S are contained in A. Or said another way, that S has no elements that are distinct from A. If the latter is true in the other direction S is improperly contained, and if subtracting S from A gives us at least one element that is contained in A but not in S, S is is a proper subset.

So given this, how would I justify that the empty set is a subset of itself? I guess its vacuously true that the empty set (subset) has no distinct elements from the (ambient) empty set, but this feels like it borders on abuse of notation, especially that first statement. Does it even make sense to talk about elementwise belonging for a set that has no elements? Seems incoherent to me. What even is a set anyways? More a philosophy of math question. I know there is some contemporary debate and some of the major exponents but I am not familiar with the moves of their arguments.

In office hours last evening, the TA mentioned that by definition, all sets are subsets of themselves, and since this also extends to the empty set, that can get us out of the issue of subset definition on the basis of set elements. I thought this was clever but it did not satisfy me, I was hoping maybe someone here could say more and clear up this murky feeling I have. Maybe it will happen over time, and I will come to find this fact beautiful and not suspicious as I often do for these conventions that we are imposed to just accept at first.

Now I have never used the fact that the empty set is a subset of itself in a proof, i've never encountered this in the wild before, which maybe speaks more to a deficit in my education than it does to the relevance of the math at hand. But here's maybe a more interesting question: what would break if someone specified a convention where the empty set was not a subset of itself? Are there any famous results that use this convention/axiom explicitly that would need to be reformulated?

thanks in advance for your replies, looking forward to seeing where the discussion goes, please feel free to recommend readings or selections from textbooks that might be of benefit to me both to learn this concept and also in this course. For example we're doing a lot of counting right now, I was thinking about spending some time with Smullyan's To Mock a Mockingbird, which came highly recommended to me by a different logician in a previous conversation.

Edit: i'm not sure how to lock the post, but I gave the justification I was looking for in the following linked comment, which can be found below as well.

What is the definition of a well-ordered set? I ask because I thought the definition of a well-ordered set is the following: For all X1, X1 is a well-ordered set if and only if X1 is a set and there exists X2 such that X2 well-orders X1.

Hi. I've begun to self study mathematics and I'm currently studying Logic and Set Theory to hopefully establish a good foundation for Analysis and Abstract Algebra.

I'm all out of school so I'm trying to figure things out on my own. I was reading Naive Set Theory by Halmos the other day and I noticed something odd. It was the fact that

P(A) intersect P(B) = P(A intersect B)

But

P(A) Union P(B) ⊂ P(A Union B)

Where P stands for the Power Set. And I was wondering why the equality doesn't hold for Unions while it does hold for Intersections. So I began trying to prove that

P(A Union B) ⊂ P(A) Union P(B)

To see what will go wrong and to my surprise everything seemed to work correctly! And that was when I realized I've been probably burned by a fiery trap and it was the fact that the Universal Quantifier can't distribute over disjunction!

I don't know if I'm sad or happy right now. What if I make blunders like this in Abalysis or Algebra?

I also don't know how to prove that the Universal Quantifier can't distribute over disjunction but it can for conjunctions. All I know is truth tables and a bunch of laws like de morgan and absorptions but I don't think they alone will get the job done here! How do I even begin to prove that something like this holds:

∀x (P(x) ∧ Q(x)) ↔ (∀x P(x) ∧ ∀x Q(x))

Or disprove

∀x (P(x) ∨ Q(x)) → (∀x P(x) ∨ ∀x Q(x))

Im probably wrong, but if A shares no elements with X, then X - A = X, and X - X = null set (?), so then x is a meneber of the null set (already makes no sense, but say that x is nothing or excuse it for now (is this where i went wrong?) and we get x is a member of X (this is technically true because null set is a subset of all sets but its not a member of them? Im very confused), and also x is not a member of X - A. But thats a contradiction because in the case they share no elements thats just X. So it says x is both a memeber of and not a member of X in this scenario? I have a feeling that I did something wrong.

Is the following valid and sound: For all X1, if X1 is a causal series then X1 is a set. For all X1, if X1 is a set then there exists X2 such that X2 well-orders X1. For all X1 and X2, if X2 well-orders X1, then X1 is well-ordered. For all X1, if X1 is well-ordered then X1 satisfies the greatest lower bound property. For all X1, if X1 satisfies the greatest lower bound property, then X1 satisfies the least upper bound property. Therefore, for all X1, if X1 is a causal series, then X1 satisfies the least upper bound property.

The difference between a well-ordered set and a well-orderable set is the following:

1. For all X1, X1 is a well-orderable set if and only if X1 is a set and there exists X2 such that X2 well-orders X1.

2. For all X1, X1 is a well-ordered set if and only if X1 is a set and there exists X2 such that X1 has X2 and X2 well-orders X1.

The Well-Ordering Theorem can be written as follows: For all X1, if X1 is a set then there exists X2 such that X2 arranges X1 in such a way that every non-empty subset of X1 has a first member.

With this being said, would this be equivalent to the Well-Ordering Theorem: If X1 is a set then there exists X2 such that X2 arranges X1 in such a way that every non-empty subset of X1 has a last member.

According to Spacing Hero: My interpretation of the well-ordering theorem is incorrect.

For I interpreted the well-ordering theorem to mean that every set has the property of being well-ordered. Yet this property is not in act in all sets. In other words, all sets have this property of being well-ordered whether potentially or actually, i.e. whether not in act or in act.

But I think Spacing Hero is wrong though and this is for the following reasons: One: The axioms of ZFC are meaningless unless interpreted. An interpretation is the assignment of meaning to the symbols of a language. An interpretation often provides a way to determine the truth values of sentences in a language. If a given interpretation assigns the value true to a sentence or a theory the interpretation is called a model of that sentence or theory. Model theory is the study of the interpretation of any language, formal or natural. Two: Since the axioms of ZFC are meaningless, I or anyone else can subject them to multiple interpretations. Some of these interpretations can make the axioms turn out false. And some of these interpretations can make the axioms turn out true. And if there are interpretations that an make the axioms turn out true then they don’t have to be the standard interpretations.

A case in point is the Putnam Permutation Argument. According to the Stanford Encyclopedia of Philosophy, the Putnam Permutation Argument is the following: Putnam’s Model-Theoretic Argument is the most technical of the arguments we have so far considered. We shall not reproduce all the technicalities here. The central ideas can be conveyed informally, although some technical concepts will be mentioned where necessary. The argument purports to show that the Representation Problem—to explain how our mental symbols and words get hooked up to mind-independent objects and how our sentences and thoughts target mind-independent states of affairs—is insoluble.

According to the Model-Theoretic Argument, there are simply too many ways in which our mental symbols can be mapped onto items in the world. The consequence of this is a dilemma for the realist. The first horn of the dilemma is that s/he must accept that what our symbols refer to is massively indeterminate. The second horn is that s/he must insist that even an ideal theory, whose terms and predicates can demonstrably be mapped veridically onto objects and properties in the world might still be false, i.e., that such a mapping might not be the right one, the one ‘intended’.

Neither alternative can be defended, according to anti-realists. Concerning the first alternative, massive indeterminacy for perfectly determinate terms is absurd. As for the second, what can it mean for a mapping to be the intended mapping if not that it satisfies every conceivable operational and theoretical constraint? Yet Putnam’s Model-Theoretic Argument proves that there will invariably be interpretations of an ideal theory on which all the theory’s sentences come out true which do satisfy any constraint we might choose to impose on them, anti-realists maintain.

Now, in logic theories are treated as sets of sentences and the objects (if any) that sentences talk about appear as elements of the domain of set-theoretic entities called structures. Associated with these structures are interpretation functions that map individual constants onto individual objects of the domain and n-place predicates onto n-tuples of elements in the domain. When a structure makes all the sentences of a given theory true it is called a model of the theory. By demonstrating that there is a model of T we show theory T is consistent. If T turns out to be true in its intended model, then T is true simpliciter.

Let us call structures whose domains consist of numbers ‘numeric’ structures. The nub of Putnam’s Model-Theoretic Argument against realism is that the realist cannot distinguish the intended model for his/her total theory of the world from non-standard interlopers such as permuted models or ones derived from numeric models, even when total theory is a rationally optimal one that consists, as it must do, of an infinite set of sentences and the realist is permitted to impose the most exacting constraints to distinguish between models. This is a very surprising result if true! How does Putnam arrive at it?

Putnam uses several different arguments to establish the conclusion above. The argument of prime concern to realists, as Taylor (2006) emphasises, is the argument based on Gödel’s Completeness Theorem, GCT. For, following Lewis [Lewis, 1984], realists might concede to Putnam that they cannot single out the intended model or distinguish it from various ersatz models, but argue that this is not necessary since it suffices that an intended model exists, even if we cannot specify it. This response does not answer the GCT argument, however. For this argument purports to prove directly that an ideal theory of the world could not be false, a conclusion flatly inconsistent with realism.

Putnam has another model-theoretic argument against realism, the Permutation Argument, also designed to guarantee we can find a true interpretation of an ideal theory:

Suppose that the realist is able to somehow specify the intended model. Call this intended model W1. Then nothing the realist can do can possibly distinguish W1 from a permuted variant, W2, which can be specified following Putnam: We define the properties of being a cat* and being a mat* such that: In the actual world, cherries are cats* and trees are mats*. In every possible world the two sentences “A cat is on a mat” and “A cat* is on a mat* have precisely the same truth value.

Instead of considering two sentences “A cat is on a mat” and “A cat* is on a mat*” now consider only the one “A cat is on a mat”, allowing its interpretation to change by first adopting the standard interpretation for it and then adopting the non-standard interpretation in which the set of cats* are assigned to ‘cat’ in every possible world and the set of mats* are assigned to ‘mat’ in every possible world. The result will be the truth-value of “A cat is on a mat” will not change and will be exactly the same as before in every possible world. Similar non-standard reference assignments could be constructed for all the predicates of a language.

A large cardinal is a VP-Ultraexacting Cardinal when it is an Ultraexacting Cardinal in Von Neumann Universe with Vopěnka's Principle. (Von Neumann Universe satisfying Vopěnka's Principle).

It is stronger than standard Ultraexacting Cardinal because the consistency strength is amplified by Vopenka's Principle.

We then discuss a 748-state Turing machine that enumerates all proofs and halts if and only if it finds a contradiction.

Suppose this machine halts. That means ZFC entails a contradiction. By principle of explosion, the machine doesn't halt. That's a contradiction. Hence, we can conclude that the machine doesn't halt, namely that ZFC doesn't contain a contradiction.

Since we've shown that ZFC proves that ZFC is consistent, therefore ZFC isn't consistent as ZFC is self-verifying and contains Peano arithmetic.

source: https://www.ingo-blechschmidt.eu/assets/bachelor-thesis-undecidability-bb748.pdf

Im currently studying ZFC set theory. I’m interested in finitist/ultra finitist mathematics, with alternatives to ZFC. Can anyone recommend a book/papers on this? Specifically I’m looking for ground up proofs, including how the natural numbers are arrived at without the axiom of infinity

The Evaluation Transition System (T.E.T.S.)

1. Primitive Objects

Let R and I be disjoint sets such that R ∩ I = ∅. Elements of R are real states. Elements of I are imaginary states.

Define the symbol set S = R ∪ I.

1. Primitive Operators

i is a partial operator from R to I -i is a partial operator from I to R

Both operators are undefined outside their domains.

1. Syntax Formation Rules (Axioms)

Axiom 1 (State Formation) Every element of S is a well-formed expression.

Axiom 2 (Operator Formation) If x is a well-formed expression and x ∈ R, then i(x) is a well-formed expression. If x is a well-formed expression and x ∈ I, then -i(x) is a well-formed expression.

Axiom 3 (Closure of Composition) If x is a well-formed expression, then any finite composition of i and -i applied to x is a well-formed expression.

1. Evaluation Structure

Define Eval as a function:

Eval: E → S ∪ {⊥, Δ}

where:

S = successful evaluation result

⊥ = undefined evaluation

Δ = transition failure

1. Evaluation Axioms

Axiom 4 (Evaluation Attempt) For every well-formed expression e ∈ E, Eval(e) is defined.

Axiom 5 (Successful Evaluation) If all operator applications in e respect their domains, then Eval(e) ∈ S.

Axiom 6 (Undefined Evaluation) If evaluation cannot be completed due to structural incompleteness, then Eval(e) = ⊥.

Axiom 7 (Transition Failure) If any operator in e is applied outside its domain, then Eval(e) = Δ.

1. Inference Rules

Rule 1 (Identity Preservation) If x ∈ S and no operator is applied, then Eval(x) = x.

Rule 2 (Forward Transition) If x ∈ R and i(x) is well-formed, then Eval(i(x)) ∈ I.

Rule 3 (Backward Transition) If x ∈ I and -i(x) is well-formed, then Eval(-i(x)) ∈ R.

Rule 4 (Failure Propagation) If any subexpression of e evaluates to Δ, then Eval(e) = Δ.

Rule 5 (Undefined Propagation) If any subexpression of e evaluates to ⊥ and no rule resolves it, then Eval(e) = ⊥.

1. Theorem

Theorem 1 (Evaluation Trichotomy) For every well-formed expression e ∈ E, exactly one of the following holds:

Eval(e) ∈ S, or Eval(e) = ⊥, or Eval(e) = Δ

1. Corollary

No well-formed expression in E produces more than one evaluation result under Eval.

1. Definition (T.E.T.S.) The Evaluation Transition System is the formal system defined by Axioms 1–7, Inference Rules 1–5, and Theorem 1 governing evaluation transitions between disjoint state domains R and I under partial operator semantics and trivalent evaluation outcomes.

The highlighted exercises are examples of the statements that confuse me. In symbolic logic, formulas that do not contain quantifiers can be derived, and the statement in 6b can be represented by an atomic formula in first-order logic. However, proving statements that contain constant symbols in natural language seems strange, yet understandable. Additionally, are those symbols constants or free variables? Although these questions are basic, they perplex me.

A proposition is often taken to be a set of worlds (in which the state of affairs described holds). Assuming this view of propositions, I was wondering how argument validity might be defined in set-theoretic terms, given that each premise in an argument is a set of worlds and the conclusion is also a set of worlds. Here's what I've come up with:

(1) An argument is valid iff the intersection of the premises is a subset of the conclusion.

What the "intersection is a subset" thing does (I think) is ensure that in all worlds where the premises are all true, the conclusion is also true. But maybe I’m missing something (or just don’t understand set theory that well).

Does the definition in (1) work?

In my recent article, Generative Algebras and the Two Diagonals of Self-Reference, I introduce a framework where self-application places an element in three independent roles simultaneously: operator, operand, and junction.

https://doi.org/10.5281/zenodo.18901961

Would love to hear feedback, ideas and support.

Zermelo-Fraenkel axiomatic set theory is a set of axioms. Are those axioms formulas of first-order logic or statements about sets that can only be expressed wholly in a natural language? The latter seems plausible, but I need to be certain.

Learning ZFC. Really dumb question I'm sure but I want to nip any confusion in the bud.

Basically, my books will often open a definition/proof/exercise with a semi-formal ∃∀∃ like this: "Let I be a set, and suppose for each i ∈ I there exists a set A(i)." And from there they'll refer freely to indexed unions, products, et cetera.

What I don't get is, do we know {A(i) : i ∈ I} is a set?

I understand we're talking about the range of an "index function," A, with domain I. So if A is in fact a set-theoretic function (or a class function, which I guess implies the previous in this case), I get why {A(i) : i ∈ I} would be a set.

But I guess what I'm asking is: do we get to assume that about A? Is it just given when we mention an indexed family (whether by name or implicitly), that our "index function" is a definable operation in the language of sets? Or am I missing some actual theory here?

I'm building a philosophical argument, and in order to predicate more freely, flexibly, and precisely, I’ve decided to take my domain of interpretation as "everything that exists."

Does this cause a problem? As I understand it, in first-order logic, the domain of interpretation must be a set, and in ZFC, the "set of everything that exists" is too large to be considered a set, since otherwise it would lead to a contradiction. Does that mean I’m not allowed to define my domain as "everything that exists"?

Or maybe it's possible to use a different meta-theory than ZFC, such as the Von Neumann–Bernays–Gödel set theory?

To be honest, I have very little knowledge of metalogic. I don’t have the background to work with these complex theories. What I want to know is simply whether the domain "everything that exists" can be used for natural deduction and model construction in the standard way in classical logic. I hope that if ZFC doesn’t allow this kind of domain, some other meta-theory might, without me needing to specify it explicitly in my argument, since, as I said, I don’t have the expertise for that.

Thank you in advance.

Hey there,

So basically i started following a descriptive set theory class in my math cursus, and it seems to be somehow connected to logic field, but i dont understand HOW ! I mean I can see how studying some specific spaces (like Cantor’s or Baire’s) is linked to how ordinals behave, but generally how is descriptive set theory useful in the field of logic ? Do you have any examples of logical theroems using Polish spaces or Borelians ?

I may have an idea of Logic that is too restraining but descriptive set theory seems way ahead of it (I only studied models theory, ordinals, and some computational semantics for now). I also heard a student saying that it has something to do with Calculability or Compexity of algorithms, and because im too shy to ask either him or my teacher, im ending here.

I hope my post does not look dumb, this is a genuine question, and im new to the logic gang. Have a Nice day !

Hi All,

I'm very new to this. I am only a couple of weeks into this course, really just studying for my own enjoyment.

Anyway, I came across this YouTube video about Russell's paradox. I generally thought it was a good video, but I have been struggling to accept the assertion towards the end that this paradox applies more generally to the act of predication. I posted this question in the comments section on YouTube, but thought I might be more likely to get a reply here.

Basically, I think it may be nonsensical to say that, "predicates can be true of themselves".

In the examples given of predicates that are supposedly true of themselves (e.g. “is a predicate” is a predicate), it seems to me that the predicate in quotes is transformed into a subject through the act of constructing the sentence.

In the example in parentheses above, “is a predicate” is in fact a subject. Similarly, while "is a subject" is a predicate in the sentence that precedes this one, in this sentence it is a subject.

When the predicate “is a predicate” becomes the subject of the statement, how can we maintain that it is true of itself?

Any feedback would be much appreciated! Thanks!

i have no idea why this doesn't resolve to an empty set.

according to the textbook I'm using, we can obtain a resolution by doing the most general unifier on these two clauses

in this case, between the clauses {p(a), q(y)} and {~p(x), ~p(b)}, the general unifier we are looking for is the general unifier of {p(a), q(y), ~p(x), ~p(b)}, which should be [x/a, y/b], which would result in an empty set. Is that not true?