Norman J Wildberger is an Associate Professor in mathematics at UNSW (University of New South Wales) in Sydney Australia, and he has a PhD in mathematics from Yale. He is well-known for his excellent educational videos on YouTube and for being an Ultrafinitist. He voices some of his objections to the concept of ‘infinity’ in his recent video (below) “MathFoundations178: The law of logical honesty and the end of infinity”.

At 27 seconds into the video he announces “It’s time for us to face the music and expel infinity from mathematics”. He goes on to consider the physical universe and argues there are no “infinite things” in our world. He claims we cannot imagine an ‘infinite’ set, we only imagine we can imagine one. If we accept the existence of finite sets, does this mean other sets must exist that are ‘not finite’? Surely not, because if we can see a visible hotel it doesn’t mean an invisible one must exist, and if we have a moveable kitten it does not mean an immovable kitten must exist.

There is an argument that sets which are not finite must be infinite. This presupposes that both finite and “infinite” sets already exist, whereupon we could say the ones that are not finite are “infinite”. Professor Wildberger says “we don’t have such a pool of sets to begin with”.

Regarding the removal of infinity from mathematics, he warns “This is an argument that has enormous implications” and “a lot of mathematics really has to be re-thought”

You can watch the video below.

I (Karma Peny) posted a YouTube comment to highlight the issue that we have no clear statement to describe what mathematics is. I said that mathematics was originally devised to solve real-world problems and that it was underpinned by real-world physics. It’s apparent generic nature can create the illusion that mathematics has its own ‘existence’ and that it is not simply a tool based on real-world physics.

I compared the axiom ‘an infinite set exists’ with an axiom ‘the god of thunder exists’. We can claim both are consistent and cannot be disproved, but both these axioms are equally worthless and irrelevant in the real-world, just as are any deductions derived using these axioms. I concluded “It is often argued that the use of ‘infinity’ in mathematics has proven to be very successful, but the successes could be despite the use of ‘infinity’ rather than because of it. I suspect we will have more clarity and even more successes if we abandon the use of non real-world axioms”.

My comment attracted a brilliant response from YouTube user ‘Amonojack A’. Here is part of this response… “So-called ‘pure math’ was born out of the idea that it might be worth developing mathematical objects and relations that correspond to no physical situation yet discovered, but could. Seems noble enough. The problem came when people failed to keep track of context. They floundered into musing about things that not only had no known physical analog, but that **couldn’t ever even conceivably** have a physical analog”.

Professor Wildberger was so impressed with these comments that he published them both in their entirety here, on his own website.

]]>A ‘sting’ is a deceptive operation designed to catch people out. For those of us who do not accept 0.999… =1, it can feel as if members of the mathematics establishment have conspired to perform a sting operation on the rest of the world.

This 0.9 recurring sting is an elaborate mathematical con trick that’s been around for centuries. It is not just one trick; it is a series of deceptions that work together to reinforce the illusion that 0.9 recurring equals 1.

One of the key stages of a good con is called ‘the convincer’. With financial cons, this involves an actual or apparent pay-off such as a small amount of money or a fake stock dividend document. The purpose of the pay-off is to convince the victim to part with bigger sums of money.

In our case, the convincer comes in two parts. These two tricks create the illusion that infinity is a valid concept in mathematics. Part 1 claims we can create an infinite object from a finite object, and part 2 claims we can convert it back again. These ‘convincer’ arguments can persuade us to accept a huge body of theory built on the apparent validity of mathematical infinity.

Parts 1 and 2 are described below. But beware, both parts are sneaky deceptions.

*If we try to convert 1/3 into a decimal, our algorithm produces an endless stream of threes, 0.333…**So we can say 1/3 equals 0.3 recurring.*

We should first consider if it is possible to represent 1/3 as a decimal. One third can be represented in its entirety as a fraction or in a base that has 3 as a factor, such as base 12, but can it be represented in base 10?

We can use short division to attempt to divide 1 by 3, but this technique has no defined end point; it cannot complete. For each digit processed the decimal result gets closer to one third but no finite number of digits will achieve equality with 1/3. After each ‘3’ digit we produce by our decimal division algorithm, there will always be a positive non-zero remainder part. In order to attain equality with 1/3, any remainder part must be processed in its entirety, but our algorithm cannot achieve this.

We are told that after ‘infinitely many’ digits the decimal will equate to one third exactly, there will be no remainder part, and there will be no more digits after infinitely many digits. But they cannot demonstrate how the algorithm can achieve infinitely many iterations. They cannot explain how the algorithm will then end. They cannot explain why there are no more terms after ‘infinitely many’ and they cannot explain why this does not form a contradiction with the statement that the decimal is endless. Importantly, they cannot show how the result can equate to 1/3.

In order to accept 1/3 becomes 0.333… in base 10 we do not need to see all the digits, but we do need to know how the conversion algorithm can end and how this will make the result equate to one third.

The first trick they use here is to assume ‘infinitely many’ is a valid concept. Then with infinitely many decimal places, it follows that any finite value in any other base can be entirely represented in base 10. But there still remains the problem that we cannot construct an infinite decimal.

So the second trick they use is to claim that proof of construction is not required. Instead of constructing the infinite decimal in its entirety, the repeating pattern is said to imply infinitely many digits, and this must equal 1/3 because it cannot equal any other number.

We can think of 0.333… as being the series 3/10 + 3/100 + 3/1000 + … and it is true we can always choose a series length to prove the sum does not equal any number that is not equal to 1/3. But it is also true that whatever length we choose, the sum cannot equal 1/3. So by the same argument that shows 0.333… cannot equal any number that is not 1/3, we can argue 0.333… cannot equal 1/3. This does make sense because the series has endless non-zero terms; hence it has no fixed value which means it is not a number at all.

There is no proof that 1/3 can be represented as a decimal, it is just assumed it can.

We can easily convert our infinite recurring decimal back into one third:

- Let x = 0.333…
- So, 10x = 3.333…
- Subtracting (a) from (b) we get 9x = 3, hence x = 3/9 = 1/3

As an extra convincer, we don’t even need to multiply by 10; other powers of 10 like 100, 1000 (and so on) all work just as well as 10.

This is the same flawed logic that is behind the most famous proof that 0.999… equals 1. It involves multiplying it by 10 and subtracting the original series, and this supposedly removes the endless part. The problem with this approach is demonstrated in the first of the two videos near the bottom of this page (from about 6 minutes in to 08:45).

Note that we can apply the same logic to the series 1+2+4+8+…

By simply multiplying it by 2 and subtracting the starting series we can supposedly remove the endless part giving the result 1+2+4+8+… = -1.

And so by * exactly* the same logic that gives 0.999… = 1, we can say 1+2+4+8+… = -1.

In both cases the logic is flawed because we are not lining up the 1^{st} terms of the series when we do the subtraction. In both cases above we are comparing the first (n+1) terms of the multiplied series with the first n terms of the starting series in order to create the illusion that the trailing terms all cancel out. We are simply choosing to line up the terms in a way that will appear to remove the trailing part of the series.

The con men will claim there are different ‘rules’ for diverging series. They expect us to blindly accept a made-up law that stipulates we cannot apply the same logic as we use for converging series.

One point to note, however, is that 1 is related to 0.999… in the same way that -1 is related to 1+2+4+8+…, it is the ‘fixed part’ of the expression for the sum to the n^{th} term (or the fixed part of the ‘partial sum expression’ if you prefer, but beware of using the word ‘partial’ as it implies there may be a full sum).

With the convincers expertly delivered, it is easy to be taken in by the main deception. Again this is not just one trick, it is several highly dubious notions supposedly backing each other up.

The argument that 0.999… equals 1 is same as the argument that 0.333… equals one third. And so if we have been swayed by the convincers then the main con trick is very close to completion.

They start by saying since 0.333… equals 1/3, we can simply multiply both sides by 3 to get 0.999… = 1. Next they perform the 10x – x ‘proof’, exactly as described for 1/3 in ‘Part 2’ above.

But we might still have some niggling doubts at the back of our minds… how and why does the algorithm end when it reaches ‘infinitely many’? How can it reach ‘infinitely many’? How can we get a fixed value without a last term? How can there be no more terms after ‘infinitely many’ when the series is supposedly endless?

They won’t say the sum mysteriously ends and inexplicably becomes equal to a fixed value after a magic number of terms, because this sounds ridiculous and farcical.

Instead they will dress-up this very same argument in a story using better sounding words. They will explain that if we examine how the partial sum varies as n increases, we can see the trend where the sum ‘converges to’ a ‘limit’ as n approaches infinity. A nice touch is that they simply define the sum to be equal to the limit, then this is true by definition (who needs proof when we can simply define things to be true!).

In reality, to examine how the partial sum varies we should find an expression for the sum to the n^{th} term. This expression will have a fixed-part (a constant) and a variable-part (containing ‘n’). Note that the ‘fixed part’ equals the value defined by mathematicians to be the ‘limit’ of the series.

Then we are being asked to accept that if the n^{th} sum diminishes as n increases, the sum somehow equates to the fixed-part of the expression (*or the ‘limit’*) after ‘infinitely many’ terms. And since there are no more terms (after ‘infinitely many’), the sum can be said to be a ‘real number’.

But if the n^{th} sum does not diminish, then we are supposed to accept it doesn’t equal the fixed-part, and the sum is not a real number.

If we still have some doubts the con men will use put-down tactics.

They will tell us to not worry about such things; we should put them to the back of our minds. Better still, we should just forget about them completely. When we become clever enough to understand infinity we will realise there are no contradictions here; everything is completely rigorous and well-defined. After all, we are not talking about *finite* objects, these are* infinite* objects, so our concerns are simply not relevant.

The epsilon-delta formulation provides the standard definition of convergence. The con men can now use this in another way to deliver a sucker punch, just to make sure they have us down and out.

In the case of 0.999… and its ‘limit’ of 1 (or of 0.333… and its ‘limit’ of 1/3) the argument goes like this:

*If we consider the sum of the first n terms of the series (called the ‘partial sum’), then we can make this sum as close to the limit of the series as we like.**It follows that there is no number that can be placed between the value of the series and the value of the limit, therefore they must be the same value.*

The problem with point (1) is that ultimately we would like the sum to equal to the limit, and this cannot be achieved.

The problem with point (2) is that it assumes the series can somehow exist in its entirety, in which case it would admittedly have a sum that equals a fixed value. It ignores the problem of constructing a so-called infinite decimal because it assumes infinite decimals are well-defined. But if they were well-defined, their definition would describe how the terms can form a completed series, and their construction would no longer be a mystery.

For the epsilon-delta argument to make sense, you have to accept that decimals can have ‘infinitely many’ digits and that these ‘infinite decimals’ do have a static/fixed values. But if a series has endless non-zero terms then it cannot have a fixed sum. It does not have a fixed value and so it is not a number. To claim it does have a fixed value forms a contradiction with the claim it has endless non-zero terms. The con men cannot resolve this contradiction so they will simply deny it exists. No clear explanation of why this is not a contradiction will ever be provided.

Like 0.333…, 0.999… is an endless series that cannot equate to any fixed value because the series does not end; it has no last term. An endless series should be treated as an endless series, not a fixed value. Like it or not, mathematics can handle such objects without having to accept the notion of ‘infinitely many’.

The two videos near the end of this article show how such objects can be handled in mathematics, without accepting infinity.

All of mathematics is procedural. We always get from one place to another via a series of steps.

If we try to add up the natural numbers, 1 + 2 + 3 + 4 + …, we follow a process. Similarly, if we attempt to add up 3/10 + 3/100 + 3/1000 +…, we follow a process.

After n steps of the process we will have added up the first n terms. Here ‘n’ is an unknown that does not have a fixed value. Once we appreciate that n is a indeterminable variable, we can appreciate that any ‘endless sum’ cannot equal a fixed value, but it can be completely described by an expression for the sum to the n^{th} term (*where n is indeterminable*).

Moreover, an endless series (summation) is completely **and uniquely** described by the expression for the sum to the n^{th} term (*where n is indeterminable*).

Mathematics can easily work with these objects as shown in the two videos found further down this page.

As we said previously, an endless series is uniquely defined by the expression for the n^{th} sum. In the case of 0.999…, the n^{th} sum is:

1 – 1/10^{n}

This has a fixed part ( 1) and a variable part ( – 1/10^{n}).

The correct way to compare a number with an endless series is to express the number in the same format as an endless series. And so we could write 1 as:

1 + 0.n * {i.e. one plus (zero times n)}*

By elevating 1 to the same structure as 0.999… we can clearly see these are not equal.

It would not make sense to try to convert the endless series to a number in order to do the comparison, as we would then be comparing just a part of the endless series, not the whole object.

The unsuspecting car buyer handed over a large amount of cash in order to purchase a particular vehicle.

After gleefully accepting the money, the salesman handed a bumper sticker to the man. “Here’s something exactly equivalent to the car you wanted, I hope you enjoy it.” he said.

The salesman pointed out to the customer that the same bumper sticker was on the car he had intended to purchase. Everything could be explained in terms of convergence and the simple matter of defining one thing to be something completely different.

What had happened here was that salesman had compared the two objects:

*Object 1: [Bumper sticker]*

*Object 2: [Bumper sticker] + [Rest of the car]*

He could have elevated the first object into a format that made it easy to compare without losing any information about either object, like this:

*Object 1: [Bumper sticker] + Zero.[Rest of the car]*

*Object 2: [Bumper sticker] + [Rest of the car]*

Then it would have been obvious that the two objects were not identical. But the car salesman adopted a different approach. He decided to ‘define’ object 2 to be equal to [Bumper sticker].

He could even ‘prove’ this was true by taking the bumper sticker off and holding it alongside the view of the car as the car drove off into the distance. The car appeared to get smaller and smaller in comparison to the the bumper sticker.

This shows that [Bumper sticker]+[Rest of the car] * converges to* [Bumper sticker].

So they must be exactly the same thing! (just as 0.999… converges to, and therefore is equal to, 1).

Below are two 10 minute videos about why 0.999… does not equal 1.

**Why 0.999… does NOT equal 1 (Part 1: The Problem)**

**Why 0.999… does NOT equal 1 (Part 2: The Solution)**

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In early 2014, a YouTube video about mathematics went viral and soon had over 1 million views.

Using simple mathematics, it claimed to prove that if you add up all the natural numbers, 1+2+3+4+…, all the way to infinity, the answer is -1/12.

It seemed like an ‘astounding’ result because you are adding all positive whole numbers, getting bigger and bigger, and the result is:

- Not big
- Not positive
- Not a whole number

The video attracted much criticism from mathematicians. The two Physicists in the video were denounced for using non rigorous methods, and they were accused of using the words ‘sum’ and ‘equals’ far too loosely. However, several more rigorous methods were known to produce the same result, and so the true meaning of this -1/12 result remained a mystery… until now.

The true meaning of this -1/12 result is explained in layman’s terms in this response video:

The truly astonishing thing revealed in this response video is that this -1/12 result is simply one huge blunder.

The mistake is one of taking a function that applies to just positive whole numbers, manipulating it in ways that bring fractions and negative numbers into play, and then interpreting the result as though it still relates to positive whole numbers.

]]>At the very foundation of modern mathematics you will find a belief in a mysterious realm that lies outside this physical world. Inside this realm all numbers from all number systems inherently exist. This abstract world has always existed and always will. And without this imagined place it is claimed we would not even be able to prove that numbers exist.

But how could this strange state of affairs have occurred? Maybe it was the desire to prove that numbers ‘exist’ rather than can be constructed that led down this extraordinary path. Should we really accept that saying numbers exist in an abstraction is somehow a valid reason to bypass the need for any physical existence?

An idea has a physical presence in our mind (via the chemistry of the brain) or where it is documented (e.g. via the ink it is written in). In this respect, abstract ideas need a physical manifestation in order to exist. Ideas might include numbers, the rules of how to create more numbers, or even the rules of a how objects behave in an abstract system.

If I imagine some cats controlling a spacecraft and landing on Mars then this idea can be said to exist in my mind. But I cannot conclude that space cats must exist on Mars simply because I can imagine it. Instead of Mars I could use an imaginary planet and call it an abstract thought, but this still does not prove that space cats or the imaginary planet actually exist themselves. Similarly, being able to think about numbers does not prove that numbers themselves exist in an abstract world. To say something can exist without any physical presence at all is a belief in the supernatural, and this is exactly what mainstream mathematics is based upon!

If I write down a big number that has never been written before, I have simply constructed it for my own use. I have not brought it into existence for everyone else to use, and I have not fetched it out of a mysterious realm containing all numbers. But mainstream mathematicians disagree. Rather than talk in terms of construction, they prefer to talk about existence as they insist all numbers already exist.

This belief (called mathematical platonism) is that with absolutely no physical manifestation at all, an infinite number of numbers (or possibly many infinities) just ‘exist’ in an abstract realm that has existed since the start of time.

Now let’s see how this concept is used. Consider this simple fundamental proof…

*Given any two different real numbers a _{0} and b we can construct another real number halfway between them. If we call this new number a_{1} then we can show how a new number (a_{2}) can be created halfway between a_{0} and a_{1}.*

This is claimed to inductively show there are an infinite number of reals between a_{0} and b.

To me, this is just two statements of fact that prove we can construct more real numbers endlessly. But most mathematicians see this as two statements of fact that apply to all the numbers between a_{0} and b. And since all the numbers already ‘exist’ (in an eternal abstraction), then it must apply to an infinite number of numbers.

Importantly, endless and infinite are not the same thing. Infinite implies you can have an infinite number of something whereas endless implies an open-ended process where you can never reach an end point, and so you can never have an infinite number of anything.

The word ‘abstract’ has been used to supposedly justify numbers can inherently exist in a realm outside of reality. And once this strange idea has been accepted, the notion that infinity must also exist is a very small step indeed.

Mathematics would be a more staid and boring subject without all the weird and wondrous things that infinity brings with it. But do we really want mystic beliefs at the heart of a discipline that is traditionally associated with logic and rigour?

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It is inconceivable that infinity might not be a valid mathematical concept. It is used extensively throughout almost all braches of mathematics. Top mathematicians have accepted and used the concept for hundreds of years. Calculus would not exist without it. The very foundation for modern mathematics is the axiomatic system known as ZFC, which guarantees the existence of at least one infinite set. On top of all this, there are obviously an infinite number of numbers and so any argument against the concept of infinity must be invalid.

I used to accept these arguments, but concerned by the many resulting paradoxes, particularly those in Hilbert’s hotel (where infinity + anything = infinity), I began to question these views. Perhaps we should have ‘endless’ or ‘open-ended’ sets instead of infinite sets where an actual infinity must exist?

Particularly concerning to me is that when the fundamental laws of mathematics break down, as in the example of Hilbert’s Hotel, these are dismissed as paradoxes. Why are they not identified as contradictions or inconsistencies that highlight a flaw in the premise that ‘infinitely many’ of something can occur?

These problem areas are not difficult to find. For example, we can show that any set of unique (non-repeating) natural numbers, excluding zero, must contain at least one number that is equal-to or greater-than the size of the set. If this rule holds, then the set of ‘all natural numbers’ must contain one or more natural numbers of infinite size. But natural numbers cannot be ‘infinite’ by definition and so we have a contradiction.

The solution is to have another rule that says there are different rules for finite and infinite sets. But no explanation is ever provided for why does (and how can) such a rule suddenly not hold when the set contains ‘infinitely many’ elements? It seems rules have been created to provide arguments for ignoring any apparent contradictions and to support the idea that infinity is a valid concept.

The concept of infinity is used extensively throughout mathematics. Behind all of this is an appreciation of infinity derived from the argument:

*As natural numbers are endless, there must be an infinite number of them.*

Initially this sounds convincing, but on closer examination the problematic nature of this statement soon becomes apparent.

One definition of infinite is ‘limitless/endless’ or ‘without bounds’ and the sequence of natural numbers is certainly limitless (within the physical constraints of any system that uses them; note that even the human mind is a physical system). If we try to construct the set of all natural numbers by extending a finite set, we fail. It can be extended indefinitely, but at every stage of the process the sequence remains finite. All we can do is assume the infinite set already exists, and not worry about its construction.

Another definition of infinite is ‘non-finite’. This implies that an endless (infinite) sequence can form a completed (non-finite) collection. Only if the collection can be completed will it be valid to use phrases like ‘an infinite number of’ and ‘infinitely many’.

The first definition implies that no end point can be reached, as the sequence is endless. But the second definition appears to contradict this as it allows a theoretical end point called infinity. Perhaps the sequence can still be ‘endless’ and be confined within a completed object? But it is difficult to see how this could occur and it cannot be shown how this can be constructed.

We can use a mathematical representation of endless recursion or iteration to prove that an infinite object does exist. By showing that these processes do not end we can claim they are not finite and thus they must be infinite. But such proofs are only valid if it is possible for a process with no defined end point to somehow occur ‘infinitely many’ times.

Proving an infinite set cannot exist is difficult or easy depending on your point of view. If the set of natural numbers was a normal (finite) set, we could simply add 1 to the highest number to generate a number not already there. This would prove by contradiction that the set does not contain all natural numbers. But as we have no insight into the structure of an infinite set, any counter proof of its existence can be rejected on the basis that normal rules do not apply to this type of object.

For example, Cantor uses a diagonal argument to prove there is no bijection (one-to-one correspondence) between the set of real numbers in the interval (0,1) to the set of natural numbers. This proof is achieved by showing that for any given set of real numbers between 0 and 1, another real number can be constructed that is in the interval but is not in the given set. However, if we extend our given set to be the whole interval (0,1), then the diagonal argument says we can construct another real that is in the interval and yet not in the interval! Therefore it could be argued that what this really proves is that any interval of real numbers cannot exist.

Consider a set containing the three numbers:

66454517

3

507

It is always possible to construct a number not already in the set by going down a diagonal and changing each digit.

For example, first we will start by forming a number from a diagonal. For our ‘units’ column we will take the ‘units’ (least significant) digit from the first number (this gives us 7). Then we get our ‘tens’ digit from the next number (we can assume a leading zero in front of the 3), and so on. In this case we get 507, which just so happens to already be in our set.

Next we change each digit in our found number. Each digit can be changed to any other digit. And so 5 could change to 9, zero could change to 1 and 7 could change to 1 giving us 911.

This method of changing each digit in a diagonal will always generate a number not already in the set for any sized set. This means literally ANY sized set of natural numbers. Therefore we can never have a completed set of ALL natural numbers as this method proves there will always be numbers not in the set.

Does the counter argument offer any insight or glimmer of an explanation as to how a set of unique natural numbers can contain more elements than the value of the largest number in the set, or how there can not be a largest number in the set? No, it simply makes the priori assumption that ‘infinitely many’ is a valid concept and it uses this to supposedly disprove our argument. It says it is true that you will create a number not in the set, but it will have ‘infinitely many’ digits. Thus it is not a natural number, ergo it should not be in the set.

Even if we accept the starting position that ‘infinitely many’ is valid, the counter argument is still self-defeating. Some digits created by our diagonal algorithm will be a finite number of digit-places away from the units-column digit, and other digits must be an infinite number of places away from the units-column digit (if we are to have an infinite result). This forms an inconsistency. This type of inconsistency should not occur because our algorithm is uniform. How and why this inconsistency occurs cannot be explained.

Is there a counter argument to this? Not really… whenever an inconsistency (or contradiction) like this is encountered, it is shrugged off and ignored by the simple act of calling it a paradox. This means it cannot be explained right now, but it will be shown to not be an inconsistency or contradiction when we have a better understanding of infinity. One very weak argument is that infinity is an axiom making it true by definition, end of story. Another equally weak argument is that infinity is well-defined resulting in entirely consistent mathematics. In other words, based on blind faith that infinity is a valid concept, and even though the concept is not understood/defined well enough to explain the many paradoxes, all counter arguments to its validity are deemed to be false.

Why do we need completed sets such as the ‘set of all integers’ or the ‘set of all natural numbers’? After all, instead of saying a variable belongs to the set of all integers we could say its ‘type’ or ‘class’ is integer, meaning it matches the description of what an integer is.

Also, infinity can create problems that otherwise would not exist. For example, if we accept the infinite set of all natural numbers exists {1, 2, 3, 4…} , it is valid to ask “what is the percentage of odd numbers in this set?”.

The obvious answer of 50% is problematic because it implies an even number of elements whereas infinity is supposedly neither odd nor even. Another option is to say the cardinality of the set of all odd natural numbers is exactly the same as the cardinality of all natural numbers, which is far from a satisfactory answer.

Without infinity we can simply say the percentage of odd numbers in the first n natural numbers is

100*floor[(n+1)/2)]/n where n>=1 * {NOTE: ‘floor’ means ‘round down’}*

Without infinity this provides a complete solution for all natural numbers and we no longer have any problems.

And so for problems like this, we could re-phrase questions to refer to ‘the sequence of natural numbers up to n, where n is unlimited’ rather than referring to ‘the set of all natural numbers’.

The use of the word infinity can be described as misleading at best. More accurate terminology should be used in its place. For example, instead of saying

*as x tends to infinity, f(x) tends to whatever*

it would be more correct to say

*as x increases, f(x) has a limit given by whatever*

The article ‘investigation of infinity in mathematics’ explains why even the simplest and most common uses of infinity are problematic. It argues that all uses of the concept should be completely removed from mathematics.

- If an endless process (of iteration/recursion) is offered as proof that an infinite object can be constructed, then a contradiction is created (as a completed infinity implies we can achieve an end)
- Despite this contradiction, showing that recursion or iteration can be endless is claimed to prove that something can be non-finite
- An infinite set cannot be constructed
- As we do not understand how it is built, we can only make assumptions about the properties of an infinite object.
- This lack of understanding makes it difficult to prove or disprove it as being a valid concept.
- Diagonal arguments are accepted when used to prove a set is not countable, but rejected when used to prove an infinite set cannot exist
- The concept of infinity adds nothing but mysticism. It leads to problems and paradoxes that do not exist without it.
- The intangible nature of infinity lacks the precision and clarity traditionally expected from mathematics

*Gauss’s views on the subject can be paraphrased as: ‘Infinity is nothing more than a figure of speech which helps us talk about limits. The notion of a completed infinity doesn’t belong in mathematics’.*

*Many mathematicians agreed with Kronecker that the completed infinite may be part of philosophy or theology, but that it has no proper place in mathematics.*

*Mayberry has noted that “The set-theoretical axioms that sustain modern mathematics are self-evident in differing degrees. One of them – indeed, the most important of them, namely Cantor’s axiom, the so-called axiom of infinity – has scarcely any claim to self-evidence at all”.*

**The first definition of infinity:** Infinity has two definitions, the first of which is *“the state or quality of being infinite”*. This leads us to look up *‘infinite’* which also has two definitions.

**The first definition of infinite:** The first definition is *“limitless or endless in space, extent, or size; impossible to measure or calculate”*. To understand the first part of this definition we can apply it to the sequence of natural numbers (1,2,3,4,…). Here we can appreciate that however high we go, we can always add 1 to get the next number, and in this sense the sequence is endless.

Unfortunately we cannot demonstrate this property of endlessness in the real world. Any counting process, either on a computer or in a human brain or implemented some other way, will inevitably hit a physical limit where it cannot count any higher. Taking this on-board, we could claim that endlessness means there is no defined end point within the implementation constraints.

The second part of this definition talks about being *‘impossible to measure or calculate’*. But if we understand what an endless sequence is then we already appreciate that trying to measure its length does not make any sense. And so on one hand this second part of the definition could be dismissed as just being superfluous, but on the other hand it could have been purposely written to imply that an infinite object can exist but cannot be measured. If so, this infers a non-finite object can exist without providing any explanation how.

**The second definition of infinite: **The second definition of infinite is *“another term for non-finite”*. For the first definition we visualised the endlessness of natural numbers. From this, we know we can never have **all** natural numbers because we can always add 1 to the largest (within implementation constraints). This second definition directly contradicts the first as its hidden implication is that we **can** gather all natural numbers together in an infinite object. And we are still left with no clue of how to construct something that is non-finite.

**The second definition of infinity:** Returning to the definitions for infinity, the second definition claims to be the one used in mathematics. It says *“a number greater than any assignable quantity or countable number (symbol ∞)”*. Most mathematicians claim infinity is not a number and yet this mathematical definition claims it is. To understand the rest of this definition we first have to understand *“assignable quantity”* and *“countable number”*.

As there does not appear to be a mathematical definition for *“assignable quantity”* I suspect that word *‘assignable’* has been used for the purpose of misdirection. Assignable or not, it is far from clear how numbers can reach a limit or how this thing called infinity can be greater than this limit.

Similarly there is no definition for a *“countable number”* but Google offers up the definition of a *“countable set”* in its place. This definition assumes the set of all natural number exists, and so even if we could use this *“countable set”* we would have a definition that already assumes an infinite object exists.

**Can we use these definitions?**

If there were just one definition for infinite, and it was *‘endless aside from implementation constraints’* then we could say with some confidence that the sequence of natural numbers was infinite. But there would be no need to use the word ‘infinite’ at all because *‘limitless’* or *‘endless’* would covey the concept with greater clarity. And as we have already seen, the concept has several definitions that make little sense. So if we describe something as infinite then we infer these multiple meanings, which can be described as unclear at best.

We could go on to discuss the meaning of *‘exist’* but, as we have failed to get a satisfactory definition for infinity, there is little point.

**And so what is the definitive answer?**

Many people insist infinity must exist. They claim it is obvious because they believe there exists* ‘an infinite number of’* numbers. But the problem remains that the definitions of the core concepts do not stand up to reasonable scrutiny.

And so the definitive answer to *“does infinity exist”* is that the question cannot be answered as ‘*infinity*‘ does not have any clear meaning.

See ‘Investigation of infinity in mathematics‘ for more about the definition of infinity.

]]>The Extreme Finitism mission has begun.

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The hope is to improve clarity and rigour within academic disciplines through the abolition of the nonsensical concept that is infinity.

]]>Mathematician Georg Cantor introduced the concept of an infinite set. He proceeded to (supposedly) prove there are more real numbers than natural numbers.

But with infinity, we can effectively prove whatever we want to. And so for a bit of fun, let’s assume we understand what infinity is and that infinity does exist. Now we can prove the exact opposite to Cantor.

Consider feeding the number Pi (=3.14159…) into a special function that converts real numbers into natural numbers. The algorithm of our special function proceeds as follows.

If the ‘real’ number is negative, our result will start with a 3, otherwise it will start with a 2. Next we have a string of 1’s, the length of which equates to the part of the real before the decimal point plus 1. For Pi, this means that we will have 3 + 1 = four 1’s. Next we have a string of zeros, the length of which equates to the part of the real after the decimal point, to N decimal places. Thus to represent the number for two decimal places we will need a string of fourteen zeros.

Thus for any value of N, our function will produce a single natural number that uniquely represents that ‘real’ to N decimal places.

This logic is true for all values of N. Therefore if we allow N to increase to infinity, it follows that our function will be able to produce a single natural number representing that real to N decimal places (where N = infinity).

This shows that all reals can be represented by natural numbers that only contain digits of 3 and below. And since we know that natural numbers exist with higher digits in them, clearly there must be more natural numbers than reals!

The usual objection to this logic is that natural numbers cannot contain ‘infinitely many’ digits. But any set of unique (non-repeating) natural numbers, excluding zero, must contain at least one number that is equal-to or greater-than the size of the set. Why does (and how can) this rule suddenly not hold when the set contains ‘infinitely many’ elements?

It seems we can pick and choose which fundamental rules of mathematics suddenly no longer apply where infinity is involved, as long as our choices support the idea that infinity is a valid concept.

All this is, of course, complete nonsense. The first mistake is the assumption that we understand infinity as we do not have a tangible definition that contains logical rigour. In short, we don’t know what we are talking about!

**Any proof or argument that involves infinity is inherently flawed because there is no sound mathematical definition for the concept of infinity.**

See ‘Investigation of infinity in mathematics’ for more about infinity.

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