Banno

Starting by misunderstanding what is at issue, and then inventing for yourself the opposing case, makes the issue very easy to decide. Well done.

Thanks. I hope nothing I've said is at odds with this? Good feedback. I suppose that while transfinite numbers are not much used in physics, continuum...

Indeed. And not just that. Much of modern maths would be unavailable or need reworking, with no apparent gain. Magnus's position appears incoherent, i...

The very first line of the proof does exactly what you ask for here. A function maps a each individual in one domain with an individual in the other. ...

This is perverse. That is exactly what has been shown. That each element of ? can be paired with an element of ??, and that each element of ?? can be ...

It's as if someone were to say "A circle is a plain figure with every point equidistant from a given point", and you were to insist that such a thing ...

Yes! The diagonal argument and its friends are amongst the most beautiful and impressive intellectual presentations. I pity those who do not see this....

Leave your mind too open, and folk will fill it with garbage.

A description close to Davidson's anomalous monism, the view that while thoughts and actions are physically grounded (monism), there are not governed ...

Yep. The proof given shows that for each element in \mathbb{N} there is exactly one element in \mathbb{N}_0 . Take any element of \mathbb{N} and there...

Well, yes. In standard mathematics, we can define a function f: ? ? ??, f(n) = n ? 1, and check the definition. We saw that every n ? ? maps to exactl...

While it's good to see you using some formal notation, this isn't an example of a function. A function from A to B is a set of ordered pairs satisfyin...

Thanks - your acknowledgement is appreciated. Excellent use of the chess analogy.

Things would be so much easier if everyone just accepted this dictum. :wink:

Cheers. It's a point that is missed in almost all the ethics hereabouts. But the argument in this thread was particularly poor.

Interesting. I'm not saying it's not true, but that it's not even true, or false. It's not well formed enough to be true or false. Some strings of wor...

Glad to have helped.

Yeah, ok, but there might be a difference in scale... Spain is quite small. The Diamantina would only just fit.

I was thinking more of rivers such as the Todd, that starts nowhere, passes through Alice Springs, ending up nowhere. Indeed, it only contains water f...

So in your scenario, it is not possible to assign Fred to one of the populations, but you maintain that the distinction is meaningful. That strikes me...

Fair, that. Many of our rivers give up before they get to the sea, not for lack of trying, but for lack of wet.

There certainly is something amiss. The Murray-Darling is absent. Is Scotland large enough to have a river? Takes the off-the-shoulder look to an extr...

Dried will do. Fresh is much better.

On the tree out the front. Drop past and pick some, anytime. An essential ingredient for stew, casserole, many sausage dishes. Laurus nobilis makes a ...

That's a group of symbols... so you mean the f? And your claim is that the definition f: \mathbb{N} \to \mathbb{N}_0, \quad f(n) = n - 1 "specifies wh...

Where do you think this claim appears in the proof? The claim “infinity + 1 = infinity” does not appear anywhere in the proofs I have used. The proof ...

Well, it's not just me... The definition you suggest cannot be used effectively with infinite sets. But enumeration, that is a surjection, will do eve...

I taught seventh grade physics. :wink:

For anyone keen on a heavy read, The Size of Sets is an Open Logic chapter that goes through most of this. It's a work in progress, so a bit patchy. I...

Taking this a step further than anyone could possible have cared about... Now we have a modal issue, since -40ºC and -40ºF are rigid designators - the...

no... -40ºC is -40ºF.

:rofl: Nice. Now I'm not sure if you did that on purpose, or not...?

& , please excuse my interjecting. How would we be able to distinguish between these two populations? Suppose Fred presents himself to your laboratory...

Gentlemen may put their jacket back on.

Yeah... good point. I overstepped. So in both classical and constructionist maths, for any number we can construct its successor. Ok. So constructivis...

I suppose so. I don't see that a constructivist would have issues with f(n)=n-1 or f(n)=n+1. Again, these are not examples with which a constructivist...

Cheers, . It appears we now agree as to almost everything. The flower has many properties, perception makes some of these - colour, smell, shape - sal...

Yes! Magnus's objections are framed as an internal problem with a proof, when they should be framed as external problems with the process being used. ...

Yep. that's what a proof does.

Here's the definition again: f: \mathbb{N} \to \mathbb{N}_0, \quad f(n) = n - 1 \quad f(n) = n - 1 as it stands is not a definition of a bijection. It...

\mathbb{N}_0? Well, no. It is defined as f(n)=n?1 and then shown to be a bijection. That definition does not mention bijectivity at all. At this stage...

Yep - not small differences. I hope. to get back to our other conversation soon. First cool day in a week so gardening to catch up with.

Well... not quite, although there are simialriteis. What's absent, amongst other things, is the usual, somewhat naive view that truth is about practic...

I think it'd be more informative to answer "Look over there... see that? it's a ship". Show, don't tell. (Edit: Notice that this is public and communa...

Well, metaphysics is just conceptual plumbing, after all. So metaphysics is "definitional". Btu yes, I'm really not advocating direct realism so much ...

We define a function: f: \mathbb{N} \to \mathbb{N}_0, \quad f(n) = n - 1 Well-defined: For every n \in \mathbb{N}, we have n \ge 1, so n-1 \ge 0. Henc...

While addressed to hypericin, this post is for all. This might be a side-issue, or perhaps the following point is worth making. There's a line of argu...

There's a parochial madness here that is pretty sad. The supposed "ideological crisis" is a result of dropping any pretensions of acting ethically, in...

So wrong, and yet in being so worng, so right.

I very nearly missed your post, yet it's the part of the discussion that I think is novel. Much of the rest has been gone over many times in these for...