Steve Patterson, among many points of objection, is doubtful on the modern concept of infinity (or more strictly the transfinite):

The foundations of modern mathematics are flawed. A logical contradiction is nestled at the very core, and it’s been there for a century.

Of all the controversial ideas I hold, this is the most radical. I disagree with nearly all professional mathematicians, and I think they’ve made an elementary error that most children would discover.

It’s about infinity. I’ve written about infinity here, here, and here, and each article points to the same conclusion:

There are no infinite sets.Not only do infinite sets not exist, but the very concept is logically contradictory – no different than “square circles”.

Infinite sets are quite literally enshrined into the modern foundations of math – with what’s called “The Axiom of Infinity”. It simply states that, “At least one infinite set exists.” Specifically, the set of natural numbers (1, 2, 3, 4, 5, and so on).

Let’s sample one of his arguments, from later down in the same post:

First, we need to define our terms. “Infinite” or “infinity” has many definitions, some better than others. I will focus on two definitions: the standard one, and then a superior one.

The standard definition of “infinite” means “never-ending”, “limitless”, or “without boundaries.”

The superior definition of “infinite” means “without inherent limitation.” These two definitions often get mushed together, and it results in conceptual confusion.

The difference between these two definitions is metaphysical, as I will explain. Take the question:

“How many positive integers are there?”

The standard response is, “There is an infinite amount” – implying that there is an “actually-infinite” amount. That somehow, you can put “all the positive integers” into a set, and the amount of elements you’ll end up with is “infinity”.

In fact, mathematicians have a term for the

actual sizeof the set of positive integers. They call it “Aleph-null.” According to modern set theory, originally conceived by Georg Cantor, Aleph-null is the smallest size of infinity. Mathematicians think there are differentactual sizesof infinite sets.This is nonsense and a confusion about the metaphysical status of numbers, which I’ll get into later. A superior response to the question, “How many positive integers are there?” is to say:

“There is no inherent limitation to the size of set you can create with positive integers.”

That doesn’t mean there’s an actually-infinite set out there in the world. It means there’s no limit to the size of the set’s construction.

Now, there are a few holes in this reasoning (hang on, we are going somewhere good!) that tie back to a key question, what is Mathematics? Not, *what does the word mean* but: *what is the substance, the essence of the discipline and what it studies*.

The best answer I have found, building on what was said by a distinguished professor long ago now, in an aside in my Uni’s good old N2 course, M100: * [the study of] the logic of structure and quantity*.

What is being missed here, is that there is a structure being laid down by going, 0,1,2 . . . k, k+1,k+2 . . . n, n+1 . . . and that it has an associated, countable scale or quantity. one, that we can label aleph null then study as a quantity in its own right. Related, we can use ordinality as a structure to develop ordinal and even transfinite numbers.

Where, we can also conceive of an extension to the number line with a value H beyond any n in the natural counting numbers, N. Then, use our favourite catapult 1/x to associate h = 1/H, number closer to 0 than 1/n for any number n we can actually complete counting to. H is a transfinite hyperreal, h is a tamed infinitesimal as was envisioned by Archimedes and co., later by Newton, Leibniz and Euler et al. It took Robinson to tame such. The hyperreals are set R*.

(NB: Notice, I am here further defining n, where n+1 is obviously also finite and BOUNDS n by succeeding it. *That is how n is finite, it is a whole number value that mileposts the reals line R and is bounded by onward integers*. **The infinite, by contrast is any quantity that cannot be bounded by at least one definite, finite value.** That is the proper sense of infinite as meaning *beyond bounds or limitless*. The limit in question, being itself finite. *There is thus no conceptual barrier here to greater and greater transfinite numbers, ordinal or cardinal*. Also, the catapulting between h and H via 1/x unifies the extended number line R*. The basic number line quantities are a unified whole and can be assigned to definite collections, i.e. sets. Namely, N,Z,Q,R,R*. Notice, I have skipped C, which is a two dimensional vector domain created from the reals and a second axis rotated by 90 degrees, the so-called imaginary numbers, in fact vectors of rotation.)

In short, there is room for transfinite numbers of scale — notice, scale, order of magnitude — aleph-null and beyond.

Back to Patterson:

Mathematicians use phrases like:

“The set of all positive even integers.”

They claim the size of that set is infinite – specifically, it is “Aleph-null”, which is the smallest infinity. Infinite sets with larger cardinalities are called “Aleph-one”, “Aleph-two”, and so on. There are, according to mathematicians, an infinite amount of sizes of infinite sets. This was the ground-breaking work of Georg Cantor, on top of which modern mathematics is built.

Now, instead of referencing “the set of all positive even integers”, imagine we’re talking about “the set of all positive odd integers.”

The cardinality, as you might intuitively think, is the same. Aleph-null.

What about the question:

“What is the cardinality of the set of all even

andodd integers together?” In other words, what is Aleph-nullplusAleph-null?The answer: Aleph-null. The cardinalities are the same.

If this strikes you as logically contradictory, that’s because it is, but mathematicians have believed this for over a century.

This means they accept the following idea: a whole can be the same size as its constituent parts, because “Aleph-null” is the same size as “Aleph-null plus Aleph-null.”

Nope, and it is not because there is an elementary error:

They justify this by saying, “Regular finite logic doesn’t apply when talking about

infinitethings!”

The real issue is of course the implications of there being no finite bound:

0, 1, 2, 3, 4, 5, 6 . . .

0, 2,4,6 . . . = 2×0, 2×1, 2×2, 2×3, . . . 2xn, . . .

1,3,5 . . . = 2×0 +1, 2×1 +1, 2×2+1 . . . 2xn+1 . . .

k, k+1,k+2 . . . = k +0, k+1, . . . –> k-k+0, k+1-k, k+2 -k . . .

That is, *once there is no finite bound involved, N can be transformed into a great many sets that have the same scale*, a quantity that can be labelled aleph null and specified as the size of N. That is, a scale such that a set can be without limit put into 1:1 correspondence with N. Logic of structure and quantity at work again, which we duly need to study.

So, the Mathematicians are quite correct, once there is not a finite bound, sets similar to the above, though seemingly smaller than N have the same scale as N. Strange, but not incoherent. We just need to accept a paradigm shift.

As usual.

This becomes even more interesting as Patterson unveils his underlying concepts:

In order to understand the refutation of Cantor’s Diagonal Argument, we have to understand the metaphysics of mathematics – what numbers are, and their relationship to our minds.

In a nutshell: numbers are concepts. They do not exist separate from our minds, nor do they exist separate of our conception of them.

Numbers (15, 2501, 56, etc.) are symbols used to represent concepts – concepts dealing with

amount,magnitude,and quantity. Those numbers are just like letters and words. When we construct a sentence out of letters, we’re arranging some visual medium in such a way that evokes concepts in the minds of the reader.The same is true in mathematics. The symbols of “+” and “-“ do not reference objective entities in the world. They are simply shorthand – a visual symbol – for a logical relation between our concepts.

See the key contrast? Namely, *“numbers are concepts. They do not exist separate from our minds, nor do they exist separate of our conception of them.”* and again, *“The symbols of “+” and “-“ do not reference objective entities in the world.”*

That is, Objective is here used to denote tangible and external to mind, what is not like that is thus deemed not objective, it is subjective; a human invention. The abstract, in his thought, is inherently subjective, don’t even mention Plato’s silly world of forms. This framing, however, is an error. One, reflective of the baneful effects of scientism and relativism.

Instead, start afresh from a basic observation: we are finite, fallible [= error prone], morally struggling, too often ill-willed and even stubborn. So, our first person perceptions, awareness, sense of location and orientation in the world, beliefs, opinions, reasoning claims, knowledge claims etc fall under this concern. So, we need warranting filters that can improve the reliability of such experiences, without falling into hyperskepticism. Which, more effectively defines objectivity:

In philosophy,

objectivityis the concept of truth independent from individual subjectivity (bias caused by one’s perception, emotions, or imagination). A proposition is considered to haveobjective truthwhen its truth conditions are met without bias caused by a sentient subject.

Yeah, that’s Wikipedia testifying against known interest. We may as well acknowledge when they get something that may easily have been ideologically loaded right.

Hence, too, the centrality of warranting filters in establishing objectivity. The objective is generally knowable because it has been adequately filtered from the error proneness involved in our first person experience subjectivity.

Notice, truth is accurate description of entities or states of affairs etc. Nothing in that, requires that we have concrete, tangible external objects such as a coconut tree. Abstracta such as numbers can be objective and manifest their presence in something as simple as clustering fingers into a three and a two then joining them as a five, illustrating how || + ||| –> |||||. That is an inherent pattern that holds in any possible world. Even, the seemingly silly world of forms has a germ of truth in it, abstracta eternally contemplated by an utterly wise necessary being; building on a point from Augustine.

Where, many abstracta can be expressed in words or other symbols and so are communicable. Warrant regarding abstracta at the core of Mathematics is like that. For example, consider von Neumann’s construction:

{} –> 0

{0} –> 1

{0,1} –> 2

. . .

{0,1.2 . . .} –> w, omega, first transfinite ordinal.

Of course, w is not a finite bound.

So, when we see from Patterson,

There is no “largest possible [natural, counting] number.” That’s not how numbers work. Any number N that you conceive of, I can always think of N+1. Does that mean that N+1 exists prior to its conception? Certainly not.

See how the underlying radically relativist constructiv-ISM was brought in? Long since, w as a definable ordinal number was brought in, and bounds any n in N that can be exceeded by an equally finite n+1. So, there is no definable last finite, f so f+1 = w. That is we have a fuzzy border zone for N, but we can identify what it takes to be a member and what would not be a member, N is a valid though transfinite set. And all of this is objective.

We therefore see that objectivity is a pivotal quantity and how warranting filters help us achieve it. This is not to denigrate our first person experience, but it allows us to address a key limitation, error proneness.

Bonus, we have a clearer vision of the transfinite and of Mathematics. **END**