An infinite hierarchy of delusions

[godelian]

2. To reinforce that looking at and discussing theories has been, and continues to be, what is called a 'complete waste of time'.

We are currently looking at the following theories:

ChatGPT: Theories in the ZFC consistency hierarchy

The ZFC consistency hierarchy refers to a ranking of formal theories based on their consistency strength, often determined by the existence of large cardinals. This hierarchy is not linear, but certain well-known theories are commonly placed in a rough order. Each theory proves the consistency of the weaker ones below it, assuming its own consistency.

Here’s a simplified version of the consistency hierarchy starting from ZFC:

Base Level

PA (Peano Arithmetic)
Consistency strength: relatively weak.
Cannot prove the consistency of ZFC.

ZFC (Zermelo-Fraenkel Set Theory with Choice)

The standard foundation of modern mathematics.
Assumes no large cardinals.

ZFC + Large Cardinal Axioms

Each step below includes a large cardinal assumption that cannot be proved in ZFC, but implies ZFC's consistency:
Increasing Consistency Strength:

ZFC + “There exists an inaccessible cardinal”
First step beyond ZFC.
Needed to construct models of ZFC.
ZFC + “There exists a Mahlo cardinal”
Stronger reflection properties.
ZFC + “There exists a weakly compact cardinal”
Stronger than Mahlo.
ZFC + “There exists a measurable cardinal”
Very strong; implies existence of non-principal ultrafilters closed under countable intersections.
ZFC + “There exists a strong cardinal”
ZFC + “There exists a supercompact cardinal”
ZFC + “There exists a huge cardinal”

Beyond Large Cardinals

ZFC + “There exists a Woodin cardinal”
Important for determinacy and descriptive set theory.
ZFC + Projective Determinacy (PD)
Implies existence of many Woodin cardinals.
Not provable in ZFC.
ZFC + AD (Axiom of Determinacy) in L(ℝ)
Incompatible with AC (Axiom of Choice), but useful in inner model theory.

Theories Beyond ZFC in Strength but Not Just by Cardinals

ZFC + “V = L” (constructible universe)
Not stronger than ZFC in consistency but has implications on cardinal structure.
ZF + DC + AD (in place of AC)
Used in descriptive set theory.
MK (Morse–Kelley set theory)
Stronger than ZFC, allows class comprehension.
ZFC + Forcing Axioms
e.g. Martin’s Maximum (MM), Proper Forcing Axiom (PFA)
Usually require the existence of supercompact cardinals.

Summary Table (simplified):
Theory Notes
PA Arithmetic base
ZFC Standard set theory
ZFC + Inaccessible Models ZFC
ZFC + Mahlo Reflection principles
ZFC + Measurable Large cardinals begin
ZFC + Supercompact Huge structure strength
ZFC + Woodin Descriptive set theory
ZFC + PD Implies many Woodins
ZFC + AD (L(ℝ)) Incompatible with AC
MK Includes proper classes
ZFC + PFA/MM High consistency strength

Would you like a graphical diagram or explanation of how certain large cardinals correspond to inner models like L, L[μ], or L(R)?

I am currently double checking if the hierarchically higher theories merely allow for the construction of a bigger model Vκ, just like ZFC+inacc, or if they do something more than that.

You see, this hierarchy operates on a very basic principle. if a theory provides a κ cardinal that is unreachable in ZFC, it can provide the domain for a model that is closed under set operations in ZFC. This domain won't be closed in the theory that provides κ, but that does not matter, because you are trying to prove the consistency of ZFC and not of that stronger theory.

Theories are essential in math. They are not a 'complete waste of time' for someone who actually likes math. If you are not interested in the consistency of ZFC, then why are you even replying in this post?

[godelian]

With 'this kind of belief' of yourself, here, there is no wonder how and why you also arrived at 'the conclusion', and belief, that 'higher set theory is a delusion'.

Well, numerous great mathematicians were finitist. They certainly thought that infinite objects are delusional:

https://en.wikipedia.org/wiki/Finitism

Finitism is a philosophy of mathematics that accepts the existence only of finite mathematical objects. It is best understood in comparison to the mainstream philosophy of mathematics where infinite mathematical objects (e.g., infinite sets) are accepted as existing.

The main idea of finitistic mathematics is not accepting the existence of infinite objects such as infinite sets. While all natural numbers are accepted as existing, the set of all natural numbers is not considered to exist as a mathematical object. Therefore quantification over infinite domains is not considered meaningful. The mathematical theory often associated with finitism is Thoralf Skolem's primitive recursive arithmetic.

The introduction of infinite mathematical objects occurred a few centuries ago when the use of infinite objects was already a controversial topic among mathematicians.

One position was the intuitionistic mathematics that was advocated by L. E. J. Brouwer, which rejected the existence of infinite objects until they are constructed.

David Hilbert, one of the greatest mathematicians of all times, even believed that it should be possible to prove the consistency (and completeness) of ZFC using finitistic means:

Another position was endorsed by David Hilbert: finite mathematical objects are concrete objects, infinite mathematical objects are ideal objects, and accepting ideal mathematical objects does not cause a problem regarding finite mathematical objects. More formally, Hilbert believed that it is possible to show that any theorem about finite mathematical objects that can be obtained using ideal infinite objects can be also obtained without them. Therefore allowing infinite mathematical objects would not cause a problem regarding finite objects. This led to Hilbert's program of proving both consistency and completeness of set theory using finitistic means as this would imply that adding ideal mathematical objects is conservative over the finitistic part.

I am perfectly fine with large cardinals and their use in higher set theory. However, I am also aware of the fact they were historically considered to be delusional. Hence, the title of my post: We typically use a hierarchy of delusions to prove the consistency of mathematics.

[Impenitent]

the papal diet plan prevents large cardinals...

-Imp

[Age]

2. To reinforce that looking at and discussing theories has been, and continues to be, what is called a 'complete waste of time'.

We are currently looking at the following theories:

ChatGPT: Theories in the ZFC consistency hierarchy

The ZFC consistency hierarchy refers to a ranking of formal theories based on their consistency strength, often determined by the existence of large cardinals. This hierarchy is not linear, but certain well-known theories are commonly placed in a rough order. Each theory proves the consistency of the weaker ones below it, assuming its own consistency.

Here’s a simplified version of the consistency hierarchy starting from ZFC:

Base Level

PA (Peano Arithmetic)
Consistency strength: relatively weak.
Cannot prove the consistency of ZFC.

ZFC (Zermelo-Fraenkel Set Theory with Choice)

The standard foundation of modern mathematics.
Assumes no large cardinals.

ZFC + Large Cardinal Axioms

Each step below includes a large cardinal assumption that cannot be proved in ZFC, but implies ZFC's consistency:
Increasing Consistency Strength:

ZFC + “There exists an inaccessible cardinal”
First step beyond ZFC.
Needed to construct models of ZFC.
ZFC + “There exists a Mahlo cardinal”
Stronger reflection properties.
ZFC + “There exists a weakly compact cardinal”
Stronger than Mahlo.
ZFC + “There exists a measurable cardinal”
Very strong; implies existence of non-principal ultrafilters closed under countable intersections.
ZFC + “There exists a strong cardinal”
ZFC + “There exists a supercompact cardinal”
ZFC + “There exists a huge cardinal”

Beyond Large Cardinals

ZFC + “There exists a Woodin cardinal”
Important for determinacy and descriptive set theory.
ZFC + Projective Determinacy (PD)
Implies existence of many Woodin cardinals.
Not provable in ZFC.
ZFC + AD (Axiom of Determinacy) in L(ℝ)
Incompatible with AC (Axiom of Choice), but useful in inner model theory.

Theories Beyond ZFC in Strength but Not Just by Cardinals

ZFC + “V = L” (constructible universe)
Not stronger than ZFC in consistency but has implications on cardinal structure.
ZF + DC + AD (in place of AC)
Used in descriptive set theory.
MK (Morse–Kelley set theory)
Stronger than ZFC, allows class comprehension.
ZFC + Forcing Axioms
e.g. Martin’s Maximum (MM), Proper Forcing Axiom (PFA)
Usually require the existence of supercompact cardinals.

Summary Table (simplified):
Theory Notes
PA Arithmetic base
ZFC Standard set theory
ZFC + Inaccessible Models ZFC
ZFC + Mahlo Reflection principles
ZFC + Measurable Large cardinals begin
ZFC + Supercompact Huge structure strength
ZFC + Woodin Descriptive set theory
ZFC + PD Implies many Woodins
ZFC + AD (L(ℝ)) Incompatible with AC
MK Includes proper classes
ZFC + PFA/MM High consistency strength

Would you like a graphical diagram or explanation of how certain large cardinals correspond to inner models like L, L[μ], or L(R)?

I am currently double checking if the hierarchically higher theories merely allow for the construction of a bigger model Vκ, just like ZFC+inacc, or if they do something more than that.

You see, this hierarchy operates on a very basic principle. if a theory provides a κ cardinal that is unreachable in ZFC, it can provide the domain for a model that is closed under set operations in ZFC. This domain won't be closed in the theory that provides κ, but that does not matter, because you are trying to prove the consistency of ZFC and not of that stronger theory.

Theories are essential in math. They are not a 'complete waste of time' for someone who actually likes math. If you are not interested in the consistency of ZFC, then why are you even replying in this post?

Again, to show and prove how and why 'looking at' and 'discussing' 'theories' is, literally, a so-called 'waste of time'.

[Age]

With 'this kind of belief' of yourself, here, there is no wonder how and why you also arrived at 'the conclusion', and belief, that 'higher set theory is a delusion'.

Well, numerous great mathematicians were finitist. They certainly thought that infinite objects are delusional:

Do you really believe that using the 'great' word, here, provides 'more weight' and/or 'more strength' to your claim and theory, here?

Is the Universe 'an object', to you?

If yes, then an 'infinite object' is an irrefutable Fact.

Once again as can be shown and proved absolutely True, Right, Accurate, and Correct.

https://en.wikipedia.org/wiki/Finitism

Finitism is a philosophy of mathematics that accepts the existence only of finite mathematical objects. It is best understood in comparison to the mainstream philosophy of mathematics where infinite mathematical objects (e.g., infinite sets) are accepted as existing.

The main idea of finitistic mathematics is not accepting the existence of infinite objects such as infinite sets. While all natural numbers are accepted as existing, the set of all natural numbers is not considered to exist as a mathematical object. Therefore quantification over infinite domains is not considered meaningful. The mathematical theory often associated with finitism is Thoralf Skolem's primitive recursive arithmetic.

The introduction of infinite mathematical objects occurred a few centuries ago when the use of infinite objects was already a controversial topic among mathematicians.

One position was the intuitionistic mathematics that was advocated by L. E. J. Brouwer, which rejected the existence of infinite objects until they are constructed.

David Hilbert, one of the greatest mathematicians of all times, even believed that it should be possible to prove the consistency (and completeness) of ZFC using finitistic means:

1. Just because you believe that a human being is 'great' and/or 'the greatest' has no bearing at all on 'the actual Truth' of what 'that human being' believes or does not believe.

2. People who believe, or disbelieve, things are, actually, not 'that great' at all.

Another position was endorsed by David Hilbert: finite mathematical objects are concrete objects, infinite mathematical objects are ideal objects, and accepting ideal mathematical objects does not cause a problem regarding finite mathematical objects. More formally, Hilbert believed that it is possible to show that any theorem about finite mathematical objects that can be obtained using ideal infinite objects can be also obtained without them. Therefore allowing infinite mathematical objects would not cause a problem regarding finite objects. This led to Hilbert's program of proving both consistency and completeness of set theory using finitistic means as this would imply that adding ideal mathematical objects is conservative over the finitistic part.

I am perfectly fine with large cardinals and their use in higher set theory. However, I am also aware of the fact they were historically considered to be delusional. Hence, the title of my post: We typically use a hierarchy of delusions to prove the consistency of mathematics.

If 'that' is what you and others do, then so be it.

[godelian]

Again, to show and prove how and why 'looking at' and 'discussing' 'theories' is, literally, a so-called 'waste of time'.

This post is about a hierarchy of mathematical theories. If you think that theories are a 'waste of time', then why do you want to engage in this post?

[wtf]

the papal diet plan prevents large cardinals...

Stan Musial is the greatest Cardinal.

[Impenitent]

Stan Laurel was funnier

-Imp

[Age]

Again, to show and prove how and why 'looking at' and 'discussing' 'theories' is, literally, a so-called 'waste of time'.

This post is about a hierarchy of mathematical theories. If you think that theories are a 'waste of time', then why do you want to engage in this post?

Once again, to, literally, show and prove how and why 'looking at' and 'discussing' 'theories' is, literally, a so-called 'waste of time'.

1. What have 'we' been doing, here, exactly?

2. When it is possible to 'look at' and 'discuss' what is actually True, Right, Accurate, and/or Correct, which 'we' could be doing, here, instead, then 'looking at' and 'discussing' what are just 'theories' is, literally, a so-called 'waste of time'. (Which is what I have been, literally, showing and proving, here.)

[godelian]

2. When it is possible to 'look at' and 'discuss' what is actually True, Right, Accurate, and/or Correct, which 'we' could be doing, here, instead, then 'looking at' and 'discussing' what are just 'theories' is, literally, a so-called 'waste of time'. (Which is what I have been, literally, showing and proving, here.)

In that case, why don't you start your own post on "what is actually True, Right, Accurate, and/or Correct" instead of derailing a post on the consistency hierarchy of ZFC-based set theories?

[Age]

2. When it is possible to 'look at' and 'discuss' what is actually True, Right, Accurate, and/or Correct, which 'we' could be doing, here, instead, then 'looking at' and 'discussing' what are just 'theories' is, literally, a so-called 'waste of time'. (Which is what I have been, literally, showing and proving, here.)

In that case, why don't you start your own post on "what is actually True, Right, Accurate, and/or Correct" instead of derailing a post on the consistency hierarchy of ZFC-based set theories?

Which are 'an infinite hierarchy of delusions', correct?

And, why I have not yet started my own post on what is actually on T.R.A.C.K is because no one, here, has yet shown any interest at all on learning, understanding, and comprehending any thing more nor anew, here.

They are too busy concentrating and focusing on 'theories', 'maybe's', or 'what if's', instead.

Oh, and by the way, you, once again, appear to have completely and utterly 'missed the point', which I just presented and highlighted above, here.

[godelian]

Which are 'an infinite hierarchy of delusions', correct?

For finitists, yes, an infinite mathematical object is a delusion.

For example, as an object, the natural numbers do not exist in finitism, because it would be an infinite object.

If you believe that it exists, however, by means of the axiom of infinity, you can use this belief to prove the consistency of arithmetic (PA).

Hence, the paradoxical outcome. In order to prove that there are no contradictions in arithmetic (PA), you need to believe in the existence of something that is often thought of as delusional.

[godelian]

Which are 'an infinite hierarchy of delusions', correct?

Yes, I came to the conclusion that ZFC+inacc is a theory that is racist towards what it considers to be inferior theories.

ChatGPT: ZFC+inacc believes (but cannot prove) that the cardinal κ is inaccessible to ZFC but not to itself. On the contrary, ZFC+inacc uses it to construct Vκ.

Yes, your summary captures an important subtlety in the metamathematics of ZFC + "κ is inaccessible" (ZFC+inacc). Let's unpack this carefully:

Inaccessible Cardinals and ZFC
A cardinal κ is (strongly) inaccessible if:
κ is uncountable.
κ is regular (i.e., cf(κ)=κ).
κ is a strong limit (i.e., 2^λ<κ for all λ<κ).

ZFC cannot prove the existence of any inaccessible cardinal, because such cardinals are "large" and transcend the strength of ZFC itself. However, we can extend ZFC by assuming the existence of such a cardinal:
ZFC+inacc: ZFC plus the axiom "there exists an inaccessible cardinal κ".

What Does “ZFC Believes κ is Inaccessible” Mean?

When you write:
"ZFC+inacc believes (but cannot prove) that the cardinal κ is inaccessible to ZFC but not to itself."

You're pointing out a relativity of inaccessibility:
κ is inaccessible from the perspective of ZFC: That is, ZFC cannot prove κ exists, but ZFC+inacc assumes it exists and has all the required properties.
But from within ZFC+inacc, we can reason about κ and construct the universe up to Vκ​.
However, κ is not inaccessible relative to ZFC+inacc, because ZFC+inacc can refer to and manipulate κκ directly—it "sees" it.
In other words, inaccessibility is a relative property: it's meaningful only with respect to a given theory.

Constructing Vκ

Once you assume κ is inaccessible, ZFC+inacc can build the rank-initial segment of the universe:
Vκ=⋃(α<κ)Vα
This structure Vκ​ satisfies all the axioms of ZFC, so it's a model of ZFC.

Thus:
ZFC+inacc constructs Vκ​ as a model of ZFC.
This is why κ is inaccessible to ZFC, but not to ZFC+inacc—ZFC+inacc uses it explicitly.

Summary of the Paradoxical Flavor

ZFC can’t prove any cardinal is inaccessible.
ZFC+inacc assumes κ is inaccessible to ZFC.
But once you’re inside ZFC+inacc, κ is not “inaccessible” anymore—it's just part of the universe.
Yet ZFC+inacc uses κ to define a model of ZFC: Vκ⊨ZFC
So, yes—ZFC+inacc "believes" κ is inaccessible to ZFC, but not inaccessible to itself. This reflects a meta-theoretic viewpoint, where theories can reference other theories' limits but not their own.

Let me know if you'd like to see how this relates to reflection principles or consistency strength.

So, ZFC+inacc strongly believes in (but cannot prove) its own superiority. ZFC+inacc firmly believes in (but cannot prove) the inferiority of ZFC. Therefore, the only difference between ZFC+inacc and ZFC, is the virulent racism of the former. So, ZFC+inacc uses its racist belief towards ZFC to construct model Vκ. Hence, it solves the underlying metamathematical problem by means of unsubstantiated racism.

[Age]

'This one' now claims that letters believe things.

So, 'the delusions', here, continue.

[godelian]

'This one' now claims that letters believe things.

So, 'the delusions', here, continue.

ZFC is an abstract "thing".

In line with Platonism, I believe that it is an object in a metaphysical reality:

Gemini AI

Mathematical Platonism is the philosophical view that mathematical objects, like numbers and sets, exist independently of human minds and are abstract, eternal, and unchanging. It suggests that when mathematicians discover mathematical truths, they are uncovering pre-existing realities rather than creating them.

Here's a more detailed explanation:

Core Beliefs:

Realism:

Mathematical Platonism asserts that mathematical statements are true or false based on whether they accurately describe these independently existing mathematical entities.

Abstractness:

These entities are considered abstract, meaning they don't exist in space or time and are not physical objects.

Independence:

Mathematical objects exist independently of human thought, language, and any physical or mental activity.

Eternality and Unchangeability:

Mathematical entities are thought to be eternal and unchanging, existing outside of the temporal flow of the universe.

So, yes, if you don't believe in Platonism, it all sounds delusional.