The fear of infinity is a form of myopia that destroys the possibility of seeing the actual infinite, even though it in its highest form has created and sustains us, and in its secondary transfinite forms occurs all around us and even inhabits our minds. Alas, Darwinists simply cannot accept such unsophisticated conclusions. But Cantor continued to ask and seek and knock with the zeal of a true pioneer. He insisted that mathematicians were explorers of an objective reality, not just the describers of the physical world.
And through his own explorations he discovered several types of infinities—countable infinities such as 1, 2, 3, etc. Since then, mathematicians have discovered many other infinities whose forms truly beggar the imagination. Yet there were a couple of infinite sets for which, no matter how he tried, Cantor was unable to categorize their size. In he put forward what is called the Continuum Hypothesis, a proposition about the sizes of these infinite sets.
Much to his dismay he never proved it, and his opponents said this failure justified all their skepticism. Their mockery drove Cantor into deep bouts of depression. He died in a sanatorium in He wrote two brief proofs that shook the foundations of both philosophy and mathematics.
He showed that we can never actually decide whether any mathematical axiom is true or not. Now that might almost sound like a self-contradictory statement: He mathematically proved that we cannot mathematically prove a statement to be true? Not quite: He mathematically proved that we cannot decide whether a statement is true.
Our logic can be exhaustively complete or it can be provably true, but it cannot be both at the same time. There will always be statements which we know to be true even though we cannot prove them to be true. Our logic will always be incomplete.
I only believe in a priori truth. Rather, it means that empirical science ultimately rests upon objective, self-evident truths that we cannot wrap our minds around—that we will never, ever be able to wrap our minds around. For, at its foundation, mathematics is objective, infinitely complex information. As Galileo Galilei had intuitively realized years earlier:. There are such profound secrets and such lofty conceptions that the night labors and the researches of hundreds and yet hundreds of the keenest minds, in investigations extending over thousands of years would not penetrate them, and the delight of the searching and finding endures forever.
Thus, although we can know axioms to be useful and consistent for all practical purposes, we do not get to be the authors of their absolute truthfulness. At least by us. Okay, back to the Continuum Hypothesis. To reject set theory because we cannot prove one axiom was not rational, for our descriptions of rationality and of nature will always be lacking.
Only someone who like the Intuitionist denies that the concepts and axioms of classical set theory have any meaning could be satisfied with such a solution, not someone who believes them to describe some well-determined reality.
Atheist philosopher Bertrand Russell had recently finished his massive Principia Mathematica , an ambitious attempt to replace all of religious belief and philosophical belief with mathematical logic. We can know such things by reasoned faith, but we do not get to be the authors of them. For that matter, we cannot be the authors of any such truth, only the believers, teachers, and preachers of it.
Our math can be reliable enough to run a global economy or to put men on the moon, but we still do not control it any more than we control the tides. In the end, the most formal exercise in knowledge is an act of faith.
The mathematician is forced to believe, absent all mathematical support, that what he is doing has any meaning whatsoever. The logician is forced to believe, absent all logical support, that what he is doing has any meaning whatsoever … Faith is the most fundamental of the mathematical tools.
And again, that foundation is also undeniably immaterial. Perhaps he realized it was a moot point. Our knowledge of the infinite reveals more than enough. Then in another mathematician, Paul Cohen, proved that it could not be proven either! Where does that leave us? The big takeaway is that everyone agrees we are exploring objective truth. As Brown University mathematics professor Philip J. Davis and University of New Mexico mathematics professor Reuben Hersh explained in , Darwinists have no other choice but to try to have it both ways:.
Most writers on the subject seem to agree that the typical working mathematician is a Platonist on weekdays and a formalist on Sundays. That is, when he is doing mathematics, he is convinced that he is dealing with an objective reality whose properties he is attempting to determine. But then, when challenged to give a philosophical account of this reality, he finds it easiest to pretend that he does not believe in it after all.
Why pretend? Why insist on a world of make-believe? Remember that Darwinism is dependent upon the presuppositions of materialism. Furthermore, the rationality of mathematics begs for an Author.
Any such attempt to do so would be irrational. Nevertheless, we can know by faith that both are true. In order for faith and reason to have a foundation, not merely from an epistemological viewpoint but also from an ontological one, there must be something that sustains it —a First Sustainer undergirding them all. There is no logic without a Logos. The opposite is despair, meaninglessness.
The materialist will argue that if we courageously embrace such despair, then we can give life meaning. Thankfully, the reverse is not true for either meaning or hope , for evil is entirely parasitical. But semantics can be just as breathtaking for the writer as it is for the mathematician.
And to the extent that we know there is both meaning and meaninglessness in life, we also know that we cannot be the authors of both. Knopf, , Fraenkel and E. Zermelo Berlin: Springer-Verlag, , As quoted in Infinity and the Mind by Rudy Rucker. Feferman et al Oxford: Oxford University Press, Feferman et al Oxford: Oxford University Press, , It is 1 followed by Googol zeros.
I can't even write down the number, because there is not enough matter in the known universe to form all the zeros:. For example, a Googolplex can be written as this power tower: That is ten to the power of 10 to the power of ,. But imagine an even bigger number like which is a Googolplexian. But none of these numbers are even close to infinity. Because they are finite, and infinity is We can sometimes use infinity like it is a number, but infinity does not behave like a real number.
Which is mathematical shorthand for " negative infinity is less than any real number, and infinity is greater than any real number". No, because we really don't know how big infinity is, so we can't say that two infinities are the same. If you continue to study this subject you will find discussions about infinite sets, and the idea of different sizes of infinity.
That subject has special names like Aleph-null how many Natural Numbers , Aleph-one and so on, which are used to measure the sizes of sets. But there are more real numbers such as Infinity is a simple idea: "endless". Most things we know have an end, but infinity does not. Hide Ads About Ads. What is Infinity? Example: in Geometry a Line has infinite length. A Line goes in both directions without end.
So a Line is actually simpler then a Ray or Line Segment.
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