[Edit 08 March 2016: Some of my views have changed or evolved since I posted this thread and responded to the posters in it. These have been added to the essay in a new section.]
This is a re-post from the mathematics section, where the question appears to have been off-topic. The title of this question is self-explanatory, and I have posted my opinions on it in a little 7,000-word essay on my website: http://www.thomaswhichello.com/?page_id=372. I'd appreciate hearing your views on them. The chain of reasoning and proofs I have devised are too long to be condensed here, although to present the crux of my belief-system in brief it is this:
First, that "Decimals are simply specialized adjectives of comparative size which define an implied natural number. They are not themselves a new or intermediate class of number, and, indeed, no such class of intermediate number between the natural ones exists."
Secondly, that, as decimals actually imply nothing more than natural numbers, therefore there cannot be any discrepancy of infinity between the set of all natural numbers and the set of all decimals.
"Let us say that I have a piece of cake, that is to say, one piece of cake, and I divide that piece of cake into four equal portions. I then take one of those portions. I now have in my hand one piece of cake—just as surely as the original is one piece of cake. I may, at this point, choose to analyze the physical size of that piece of cake in my hand by comparison with the piece of cake from which it was taken. This is 1/4th of it expressed as a fraction, or 0.25 of it expressed as a decimal. And yet this new piece of cake that I hold in my hand remains, numerically, as I say, one piece of cake. Yet what, therefore, does the fraction or decimal represent? What it represents is, that this particular piece of cake that I hold in my hand, although numerically one, is worth, in point of its physical size, one piece of cake out of the original piece of cake from which it was taken, once that original piece of cake has been divided into four equal portions. That new piece of cake which I hold in my hand, the decimal or fraction explains, is so much smaller in point of size as compared with that other, original piece of cake. But numerically, they are both one—both the piece I have taken, and the original which I divided up before I took it from it.
A further physical example that may more accurately demonstrate how the decimal or the fraction functions in the abstract realm, is if the reader first imagines me to have a piece of cake all on its own; secondly imagines another, larger piece of cake; and lastly is given to understand that the piece of cake I have in my hand, by virtue of sheer coincidence, is worth, in point of size, exactly one quarter (or what have you) of the larger piece of cake, though I had not taken it from that cake—simply by comparing the one thing with the other thing, and not by taking it out of it. And in this case, again, we might describe that piece of cake in my hand as being worth 1/4th, or 0.25, of the other, larger piece. This analogy, besides being more accurate than the former in its relation to pure mathematics, highlights the absurdity of calling a fraction or a decimal a number even more greatly. That I choose to compare the size of an object in my hand with that of another object does not affect the numerosity of my object, the fact that I continue to hold one object; and that is all that a decimal or a fraction does."