Does Infinity Exist?

The position that this work (SMN / IST) takes is that quantities can be arbitrarily large but they must always have a definite finite value. This was the original mathematical position too but over time the concept of infinity has been usefully applied in mathematics so the question of its actual existence has been largely ignored in favour of a pragmatic approach. In the 19th century the consensus between mathematicians began to shift and now in the 20th century there is wide scale belief in the existence of actual infinities although still no proof, indeed all the concrete evidence indicates that they cannot exist but still it is useful in mathematics so mathematicians retain their belief in its existence. This would suggest that mathematicians should check their axioms but such a reformation of modern mathematics is unlikely to occur unless there is a compelling need to do so because the logistics and the upheaval of such a reformation would be vast and it would be disruptive to many careers and traditions.

Following comments quoted from (Infinity: You Can't Get There From Here):

"Since no sensible magnitude is infinite, it is impossible to exceed every assigned magnitude; for if it were possible there would be something bigger than the heavens." (Aristotle)

Aristotle distinguished between the potential infinite, and the actual infinite. The natural numbers, he would say, are potentially infinite because they have no greatest member. However, he would not allow that they are actually infinite, as he believed it impossible to imagine the entire collection of natural numbers as a completed thing. He taught that only the potential infinite is permissible to thought, since any notion of the actual infinite is not “sensible.” So great was Aristotle's influence that more than 2,000 years later we find the great mathematician Karl Friederich Gauss admonishing a colleague,

"As to your proof, I must protest most vehemently against your use of the infinite as something consumated, as this is never permitted in mathematics. The infinite is but a figure of speech . . . ." (Karl Friederich Gauss)

Nonetheless, long before Gauss's time, cracks had begun to appear in the Aristotelian doctrine. Galileo (b. 1564) had given the matter much thought, and noticed the following curious fact: if you take the set of natural numbers and remove exactly half of them, the remainder is as large a set as it was before. This can be seen, for example, by removing all the odd numbers from the set, so that only the even numbers remain. By then pairing every natural number n with the even number 2n, we see that the set of even numbers is equinumerous with the set of all natural numbers. Galileo had hit upon the very principle by which mathematicians in our day actually define the notion of infinite set, but to him it was too outlandish a result to warrant further study. He considered it a paradox, and “Galileo's Paradox” it has been called ever since. As the modern study of mathematics came into full bloom during the seventeenth and eighteenth centuries, more and more mathematicians began to sneak the notion of an actual infinity into their arguments, occasionally provoking a backlash from more rigorous colleagues (like Gauss).

John Wallis The English mathematician John Wallis (b. 1616), was the first to introduce the “love knot” or “lazy eight” symbol for infinity that we use today, in his treatise Arithmetica infinitorum, published in 1665. Ten years later, Isaac Newton in England and Gottfried Leibnitz in Germany (working independently) began their development of the calculus, which involved techniques that all but demanded the admission of actual infinities. Newton side-stepped the issue by introducing an obscure notion called “fluxions,” the precise nature of which was never made clear. Later he changed the terminology to “the ultimate ratio of evanescent increments”. The discovery of the calculus opened the way to the study of mathematical analysis, in which the issue of actual infinities becomes very difficult indeed to avoid. All through the nineteenth century, mathematicians struggled to preserve the Aristotelian doctrine, while still finding ways to justify the marvelous discoveries which their investigations forced upon them.

Finally, in the early 1870's, an ambitious young Russian/German mathematician named Georg Cantor upset the applecart completely. He had been studying the nature of something called trigonometric series, and had already published two papers on the topic. His results, however, depended heavily on certain assumptions about the nature of real numbers. Cantor pursued these ideas further, publishing, in 1874, a paper titled, On a Property of the System of all the Real Algebraic Numbers. With this paper, the field of set theory was born, and mathematics was changed forever. Cantor completely contradicted the Aristotelian doctrine proscribing actual, “completed” infinities, and for his boldness he was rewarded with a lifetime of controversy, including condemnation by many of the most influential mathematicians of his time. This reaction stifled his career and may ultimately have destroyed his mental health. It also, however, gained him a prominent and respected place in the history of mathematics, for his ideas were ultimately vindicated, and they now form the very foundation of contemporary mathematics.

"One can without qualification say that the transfinite numbers stand or fall with the infinite irrationals; their inmost essence is the same, for these are definitely laid out instances or modifications of the actual infinite." (Georg Cantor)

Cantors definition of an infinite set was: "A set is infinite if we can remove some of its elements without reducing its size."

End of quotes from (Infinity: You Can't Get There From Here)

The article goes on to explain cantors set theory and cardinal numbers but the gist of it is "To a present day mathematician, infinity is both a tool for daily use in his or her work, and a vast and intricate landscape demanding to be explored." (Infinity: You Can't Get There From Here)

Following comments quoted from (Infinity: The Encyclopedia of Astrobiology Astronomy and Spaceflight):

By confining their attention to potential infinity, mathematicians were able to address and develop crucial concepts such as those of infinite series, limit, and infinitesimals, and so arrive at the calculus, without having to grant that infinity itself was a mathematical object. Yet as early as the Middle Ages certain paradoxes and puzzles arose, which suggested that actual infinity was not an issue to be easily dismissed. These puzzles stem from the principle that it is possible to pair off, or put in one-to-one correspondence, all the members of one collection of objects with all those of another of equal size. Applied to indefinitely large collections, however, this principle seemed to flout a commonsense idea first expressed by Euclid: the whole is always greater than any of its parts. For instance, it appeared possible to pair off all the positive integers with only those that are even: 1 with 2, 2 with 4, 3 with 6, and so on, despite the fact that positive integers also include odd numbers. Galileo, in considering such a problem, was the first to show a more enlightened attitude toward the infinite when he proposed that "infinity should obey a different arithmetic than finite numbers." Much later, David Hilbert offered a striking illustration of how weird the arithmetic of the endless can get.

Imagine, said Hilbert, a hotel with an infinite number of rooms. In the usual kind of hotel, with finite accommodation, no more guests can be squeezed in once all the rooms are full. But "Hilbert's Grand Hotel" is dramatically different. If the guest occupying room 1 moves to room 2, the occupant of room 2 moves to room 3, and so on, all the way down the line, a newcomer can be placed in room 1. In fact, space can be made for an infinite number of new clients by moving the occupants of rooms 1, 2, 3, etc, to rooms 2, 4, 6, etc, thus freeing up all the odd-numbered rooms. Even if an infinite number of coaches were to arrive each carrying an infinite number of passengers, no one would have to be turned away: first the odd-numbered rooms would be emptied as above, then the first coach's load would be put in rooms 3n for n = 1, 2, 3, ..., the second coach's load in rooms 5n for n = 1, 2, ..., and so on; in general, the people aboard coach number i would empty into rooms pn where p is the (i+1)th prime number.

Such is the looking-glass world that opens up once the reality of sets of numbers with infinitely many elements is accepted. That was a crucial issue facing mathematicians in the late nineteenth century: Were they prepared to embrace actual infinity as a number? Most were still aligned with Aristotle and Gauss in opposing the idea. But a few, including Richard Dedekind and, above all, Georg Cantor, realized that the time had come to put the concept of infinite sets on a firm logical foundation.

Cantor accepted that the well-known pairing-off principle, used to determine if two finite sets are equal, is just as applicable to infinite sets. It followed that there really are just as many even positive integers as there are positive integers altogether. This was no paradox, he realized, but the defining property of infinite sets: the whole is no bigger than some of its parts. He went on to show that the set of all positive integers, 1, 2, 3, ..., contains precisely as many members – that is, has the same cardinal number or cardinality – as the set of all rational numbers (numbers that can be written in the form p/q, where p and q are integers). He called this infinite cardinal number aleph-null, "aleph" being the first letter of the Hebrew alphabet. He then demonstrated, using what has become known as Cantor's theorem, that there is a hierarchy of infinities of which aleph-null is the smallest. Essentially, he proved that the cardinal number of all the subsets – the different ways of arranging the elements – of a set of size aleph-null is a bigger form of infinity, which he called aleph-one. Similarly, the cardinality of the set of subsets of aleph-one is a still bigger infinity, known as aleph-two. And so on, indefinitely, leading to an infinite number of different infinities.

Cantor believed that aleph-one was identical with the total number of mathematical points on a line, which, astonishingly, he found was the same as the number of points on a plane or in any higher n-dimensional space. This infinity of spatial points, known as the power of the continuum, c, is the set of all real numbers (all rational numbers plus all irrational numbers). Cantor's continuum hypothesis asserts that c = aleph-one, which is equivalent to saying that there is no infinite set with a cardinality between that of the integers and the reals. Yet, despite much effort, Cantor was never able to prove or disprove his continuum hypothesis. We now know why – and it strikes to the very foundations of mathematics.

In the 1930s, Kurt Gödel showed that it is impossible to disprove the continuum hypothesis from the standard axioms of set theory. Three decades later, Paul Cohen showed that it cannot be proven from those same axioms either. Such a situation had been on the cards ever since the emergence of Gödel's incompleteness theorem. But the independence of the continuum hypothesis was still unsettling because it was the first concrete example of an important question that provably could not be decided either way from the universally-accepted system of axioms on which most of mathematics is built.

Currently, the preference among mathematicians is to regard the Continuum Hypothesis as being false, simply because of the usefulness of the results that can be derived this way. As for the nature of the various types of infinities and the very existence of infinite sets, these depend crucially on what number theory is being used. Different axioms and rules lead to different answers to the question what lies beyond all the integers? This can make it difficult or even meaningless to compare the various types of infinities that arise and to determine their relative size, although within any given number system the infinities can usually be put into a clear order. Certain extended number systems, such as the surreal numbers, incorporate both the ordinary (finite) numbers and a diversity of infinite numbers. However, whatever number system is chosen, there will inevitably be inaccessible infinities – infinities that are larger than any of those the system is capable of producing.

End of quotes from (Infinity: The Encyclopedia of Astrobiology Astronomy and Spaceflight)

What these quotes state is that there is a definite structure of different kinds of infinities and these are useful in mathematics. But does infinity actually exist as a realisable phenomenon or is it just a useful mathematical tool? Or is it that if certain axioms are accepted then certain types of infinity arise but these are only applicable within certain axiomatic contexts. Is mathematics just exploiting the potential infinities and exploring their axiomatic structure or are there actual infinities that exist in reality and that we might someday discover. So far no actual infinities have been found and all the evidence suggests that an actual infinity is impossible. Quantum physics relies on things being quantised so there is no infinite resolution or continuum and general relativity relies on finite maximum values for velocity, mass, energy and so on so there are no objects with infinite velocity, mass, energy, etc.

It seems likely that only in the realm of pure mathematics can the idea of infinity be entertained. In the context of actual, manifest, realisable quantities things seem much more like the situation in a computer where all phenomena have definite resolution and size. One can never create an infinitely large file because that would require an infinite amount of time and infinite computational resources such as memory.

In my own work, which uses computational concepts to model reality I take the position that the phenomena can be arbitrarily large and detailed but they always have a definite finite value. So this allows for potential infinity but totally disallows actual infinity. Given that any set must be actually represented using data (e.g. binary data), then no set can be infinitely large and if one removes any members of the set then the cardinality (size) of the set is reduced. So any representable set cannot be an infinite set and any infinite set cannot be actually represented. Furthermore, in the context of computational metaphysics, representation is equivalent to existence. If something is represented and it takes part in the overall simulation of the universe then it exists in that universe but if it cannot be represented then it cannot exist. So if actual infinities exist then there cannot be any discrete computational foundation to reality but so far no actual infinites have ever been discovered.

Even with the domain of pure mathematics, infinities can only exist because they are symbolically represented and never actually represented. No one has ever written out an infinite number of integers thereby actually representing the set of integers. It is only ever referred to but never fully represented. If one required sets to be fully represented then mathematics could not operate on actual infinite sets; it could only operate on potentially infinite sets which always have finite representations (e.g. {1,2,3}) but which are unlimited in their length. Such sets are arbitrarily large but always have a definite finite size.

Modern mathematics is totally dependent upon the assumed existence of actual infinities but the existence of these can neither be proven nor disproven by mathematics. This leaves modern mathematicians in the position of defending their belief in the existence of actual infinities and discrediting any opposing ideas, but beside all of this - do actual infinites exist???? All the evidence seems to suggest that they only exist within the context of modern mathematics, which would seem to suggest that modern mathematics should return to its axioms and see where the problem arises.