On considering the comparison of cardinalities of the set of natural numbers and real numbers, we turn to Cantor’s Diagonal Argument and Cantor’s supposed proof that there exist more real numbers than natural numbers. In this essay I will firstly outline this argument and continue by setting out some of its implications. I next consider Wittgenstein and his remarks on Cantor’s argument, namely the abstract nature of transfinite numbers, the use of the term infinite and the assumption that all sets may be well ordered. Finally I will conclude that whilst Wittgenstein considers Cantor’s argument to exhibit some merit, there are fundamental flaws in these concepts which prohibit one from wholly accepting Cantor’s conclusion. To outline Cantor’s …show more content…
This number is therefore not an element of the set of the natural numbers. Thus the argument suggests there exist more real than natural numbers even when considering an infinite list of the naturals. Hence the result is given that the real numbers cannot be put into one-one correspondence with the natural numbers. This is that the set of real numbers is non-denumerable. Since we have the definition that two sets have the same cardinality if and only if there is a one-one correspondence between their elements, Cantor believed that he had shown that there is variation between the cardinality of the set of natural numbers and the set of real numbers . Defining the transfinite cardinal of the natural numbers as ℵ_0, Cantor concluded that the different cardinality of the real numbers suggests the existence of another transfinite number, larger than ℵ_0.Cantor proved the theorem 2^(ℵ_0 )>ℵ_0 stating that 2^(ℵ_0 )=ℵ_1 which Cantor defined as the cardinality of the set of the real numbers. Further, based on this idea, Cantor gave that in fact there exists an infinite number of transfinite cardinal numbers. Thus there exists an infinite number of …show more content…
Despite the supposed proof of Cantor’s theorem, Wittgenstein commented on the use of this statement in accordance with the transfinite numbers. In his Remarks on the Foundation of Mathematics, Wittgenstein noted that whilst Cantor may say 2^(ℵ_0 )>ℵ_0 it is “a piece of mathematical architecture which hangs in the air, and looks as if it were, let us say, an architrave, but not supported by anything and supporting nothing” . Although we may consider Cantor’s theorem, it in fact does not inform us as to the context of 2^(ℵ_0 ). Hence we have no more information as to this apparent cardinality of the real numbers. With no concept as to what sense or concept we are actually giving to this cardinality, we can say very little about the number of elements in the set of real numbers. Therefore to suggest conclusively that there are more real numbers than there are natural numbers, with no account for what this means, seems ambiguous and as a result
Aquinas’ argument has a couple of flaws in it. One is pointed out by Samuel Clarke, who says a whole series of dependant...
One of the most argued topics throughout human history is whether or not God exists. It is argued frequently because there are several different reasonings and sub arguments in this main argument. People who believe God exists argue how God acts and whether there is one or several. People who do not believe God exists argue how the universe became into existence or if it has just always existed. In this paper, I will describe Craig's argument for the existence of God and defend Craig's argument.
philosophy, deal with the subtleties of infinite numbers and the fact that exist not just one, but at least two and potentially an
In Descartes’s meditations, people point out that Cartesian Circle exists. However, although the argument for Cartesian Circle seems to be true, I believe this not to be the case. In this essay, I am going to first introduce the Cartesian Circle in Descartes’s argument, and then try to show why the circular reasoning is actually not what it appears to be. In the third meditation, Descartes uses the claim “whatever I perceive very clearly and distinctly is true” as a premise to prove the existence of non-deceiving God (Descartes 24).
Irvin, Andrew. "Bertand Russell." Stanford Encyclopedia of Philosophy. N.p., 18 Oct 2013. Web. 22 Mar 2014. .
My professor told us that we can comprehend God through finite numbers even though he is infinite. I have always wondered how. After reading more of this book I figured out how. Some infinite sets are tremendously larger than others and how Dr. Bessey explains it, he says, “We have already shown that the power set of {1, 2, 3} contains 23 or 8 subsets. Using the general formula, we conclude that the power set of {1, 2, 3, 4} contains 24 or 16 subsets; the power set of {1, 2, 3, 4, 5, 6, 7, 8, 9, 10} contains 210 or 1,024 subsets; and the power set of {1, 2, 3, 4, . . . , 100} contains 2100 or 1,267,650,600,228,229,401,496,703,205,376 subsets, which is more than one million trillion trillion. Exploring further, we discover that the power set of {1, 2, 3, 4, . . . , 1000} contains 21000 subsets, where 21000 equals a number that has 302 digits! Thus, by means of the power set, any finite set can be used as a stepping-stone to build another, much larger, finite set.” It is so crazy to me to think that we can understand infinity and even eternity by using finite numbers. Not only that but we can understand the eternities by understanding dimensions as
Wittgenstein, Ludwig; G. E. M. Anscombe, P.M.S. Hacker and Joachim Schulte (eds. and trans.). Philosophical Investigations. 4th edition, Oxford: Wiley-Blackwell, 2009. Print.
Although their methods and reasoning contrasted one another, both philosophers methodically argued to come to a solid, irrefutable proof of God, which was a subject of great uncertainty and skepticism. Through Three Dialogues Between Hylas and Philonous and Discourse on Method and Meditations on First Philosophy, Descartes and Berkeley paved the way towards an age of confidence and faith in the truth of God’s perfect existence actively influencing the lives of
And yet...isn't this exactly what Anselm indicated? Anselm tells us that God is "something-than-which-nothing-greater-can-be-thought". Isn't this what I've begun to describe above? I don't think so. I really only described the square circle, the empty solid, the loud silence-- contradictions, one and all. Yet, if I can't imagine a square circle or a loud silence (or any other nonsensical oxymoron), how can I possibly imagine Anselm's vision of God? Once I remove the nonsense of colorless green ideas sleeping furiously, everything I can actually imagine can reasonably be imagined +1 greater this afternoon, or tomorrow morning, or by my more imaginative neighbor. It's like counting natural numbers: you can never imagine the greatest number, because there is always another number after that one. I see God the same way--whatever I can possibly imagine is less than the sum of God; and no, saying that God is more than I can imagine is not equivalent to saying that God is more than can be imagined--and it certainly doesn't in any rational way allow the reification of
It seemed that the arguments put forward made some sense, and Charles did not or could not think of any arguments to counter. Yet, he thought the proposition was wrong; he mentioned “I am just trying to imagine what Kant would make of this.” and “It just doesn’t seem right.” Although Charles could not put a finger on it, he intuitively thought it was wrong. What appeare...
Anselm's second contention guarantees that God is endless, boundless, by or in time and in this way has important presence and is an essential being. Anselm contends that it is smarter to be a vital being than an unexpected being, a being that relies upon different things for its reality i.e. having a reason/end since this would eventually constrain your energy. He clarifies that God must be an important being on account of if God exists as an unexpected being we could envision more noteworthy, hence God would not be that than which no more noteworthy can be considered. A being which can't be imagined not to exist must be more prominent than one that can be considered not to exist. Anselm at that point clarifies it would be a self logical inconsistency
This paper's purpose is to prove the existence of God. There are ten main reasons that are presented in this paper that show the actuality of God. It also shows counter-arguments to the competing positions (the presence of evil). It also gives anticipatory responses to possible objections to the thesis.
...trass saw these problems form a purely mathematical point of view and that helped them redefine the mathematical concept of a limit. Others have thought of these paradoxes as a way of feeding our skepticism and doubting the deficiency of what we presume.
The era of his college years was also in interesting period in the realm of mathematics. Many things were and had been already changing. Mathematics was finding itself and it seemed that its rules could be fully used to find the solution to any problem. Unfortunately a known mathematician by the name of Kurt “Gödel, had proven that the axioms of mathematics never could be complete as well as consistent. This was a hard blow to many mathematicians that had been convinced that mathematics was a universal and complete system.” 2 And furthermore, there remained the question of decidability, that is, whether there was a method for deciding a mathematical statement to be provable or not.
...l. LXXIV of Creed’s journal. And taking that and what he knew he came up with “The way in which irrational numbers are usually introduced is connected with the concept of extensive magnitude and explains number as the results of the measurements of one such magnitude by another of the same kind. Instead I demand that arithmetic shall be developed out of itself (Dedekind 6).