The Sci Phi Show

Sorry,

Seems life got on top of me this week, and there will be no sci phi show this week.

Next time, I will be doing, The Dark Knight’s Joker as Nietzschean Superman

It should be fun

Infinity is the topic for Episode #34of the Sci Phi Show. Thanks for tuning in. Infinity turns up in lots of places in science fiction so I thought it would be good to overview the concept.

Podcast: Play in new window | Download (Duration: 24:33 — 12.5MB)

Show Notes

• Noodle MX the network I am affiliating with.
• Are you Just Watching?
• Wikipedia contains a good introduction to the concept of Infinity as well as an exploration of the Infinite Set and the Philosophy of Infinity
• There are also good sections on Countable and Uncountable infinities
• The Stanford Encyclopedia of Philosophy has an excellent article on Zero’s Paradoxes
• Here is a good introduction to Hilbert’s Hotel and William Lane Craig’s original paper on The Kalam Cosmological Argument where I first encountered it
• Wikipedia also has a good article on The Infinite Monkey Theorem
• Finally you can learn more about Boltzmann Brains

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Infinity to the Mathematician by Cai Wingfield

What does “infinity” mean to a mathematician?  The word “infinity” can mean one of several fairly distinct ideas in mathematics.  It usually is taken to mean something similar to “ultimate limit”, but even this doesn’t capture the full extent of it.  Let’s consider a few ways in which the word “infinity” might be used in mathematics.

The first time people come across something called “infinity”, it’s probably in the playground.

“What’s the biggest number you can count to?”
“I can count to a million!”
“Oh yeah? I can count to a million and one!”
“Oh yeah? I bet I can count to a million billion trillion!”
“Oh yeah? Well I can count to a million billion trillion and one!”
“Fine! I can count to infinity!”

This is what most people think of infinity as — some mysterious “biggest number”.  But this is not the case.  There can be no biggest number.  By definition, you can always add 1 to a number and get a bigger one.  ”Infinity” is not a number.

Let’s look a little deeper at this.  Counting numbers are things which we’re all familiar with.  However, just being familiar with something is not good enough to study it rigorously.  In order to do this, we must construct a formal mathematical object which “models” or “represents” the concept we wish to study.  What mathematicians have done is attempt to formalize the notion of a “counting number” into what they call a “natural number”.  One way mathematicians like to formalize things is by coming up with a list of “axioms” — rules which must be followed.  Then, a natural number is anything which satisfies the natural number axioms.  There are many axiomatisations of the natural numbers, but the most commonly used ones are known as the “Peano Axioms”.  They include things like “0 is a natural number”, “each natural number has a successor”, “there is no natural number whose successor is 0″, “two natural numbers are equal if they have the same successor”.  There are several more axioms, but we can see that they’re all things we’d want a mathematical account of counting number to have.

One of the more complicated axioms is “if 0 satisfies a property P and if every time n satisfies P we also have n+1 satisfies P, then P is satisfied by every natural number”.  This is called the “induction axiom”.

But now we can see that, from this definition of natural number, that “infinity” is not one.  Were ∞ a natural number, it would surely be the largest one.  We can show, however, that every natural number is distinct from its successor.  But then we can’t have ∞+1=∞, so ∞ isn’t a natural number.

Sometimes mathematicians will define something to be called “infinity” because it behaves how we’d like infinity to behave.  For example, here’s a definition from basic real analysis.

“A sequence is said to “tend to infinity” if, for each real number K, there’s a position in the sequence after which every sequent is greater than K.”

From this definition, the sequence 1, 2, 3, … “tends to infinity”, and the word “infinity” in this context is a defined term.  This does not mean that “infinity” is a number towards which the sequence tends, it merely means that “infinity” is a word used to encapsulate this concept.

This shows that we need to be extremely careful when dealing with infinity, both as a mathematical and as a philosophical concept.  Sometimes we may have an idea in our heads, like “infinity is the largest number”, but when we really sit down and study it, we realize it’s not a well-formed idea.

These two uses of “infinity”, one which makes sense and one which doesn’t seem extremely similar.  We must be very careful.  Here’s a final example which should demonstrate exactly how counter-intuitive mathematical notions of infinity can be.  It’s a story called “Hilbert’s Hotel”, after the mathematician David Hilbert.

Before we hear the story, we need a new notion of infinity.  Again, it’s very similar to our first non-example.  We’ve seen that ∞ is not a number.  But we can still ask the question “how many natural numbers are there?”  The answer to this question is, confusingly, “infinitely many”.  How can this be?  Let’s just say that while the “sizes” of sets containing finite quantities of elements correspond to the natural numbers, we can also consider the “sizes” of sets containing non-finite quantities of elements.  A set is said to be “countably infinite” when it is the same size as the set of natural numbers.

So, here’s the paradox of Hilbert’s Hotel:

Consider a Hotel with infinitely many rooms.  That is, there is one room for each natural number, and there are numbers on all the doors.

Let’s say you go to the Hotel one night to look for a room.  You’re told by the concierge:

“Sorry, the Hotel is full. There are no free rooms!”

You’re about to turn around and leave when he interrupts:

“Don’t worry, I’m sure we’ll sort something out.”

He calls the porter and says”

“Go to room 1 and kindly ask the guests there to pack up their things and move to room 2.  While they’re getting ready, go to room 2 and ask them to move to room 3.  Likewise, go to each room n and ask the guests to move to room n+1.”

He turns to you and says:

“There we go, in a few minutes we’ll have room 1 available for you.”

Amazing, you think.  You walk through the lobby and come to a long corridor.  All the doors on the left are marked 1, 3, 5, … and all the doors on the right are marked 2, 4, 6, …  Must take people a long time to walk to their room, you think.

The door opens behind you and a long queue of people trail in.  They say

“Hi, we’re infinitely-many tourists and we’d like to stay the night.”

“Sorry, the last room’s just been taken — we’re full.  Don’t worry, though, we’ll find space.”

He calls the porter again.

“Please go to each room n and tell the guests to kindly move their things to room 2n.  This will be ok for everyone.”

He turns back to the queue.

“Now all the odd-numbered rooms are free, and there are infinitely many of those, so you’ll definitely all fit in.”

“Absurd!” You say to the concierge. “Is there any limit to the number of people you can fit in the hotel?”

“Not really” he says. “Only the other day, infinitely many coaches, each containing infinitely many passengers turned up and we still managed to fit them all in!”

You’re just walking back to your new room when you notice the porter putting a newspaper on the floor of each door.  He’s just concentrating on the left-hand-side of the corridor first, putting a newspaper at each door.

“If you’re just doing the left-hand-side first, won’t it take you an infinite amount of time?  How will you ever get to any of the even-numbered doors?”

“Good point!” he says.  ”At this rate, half of the guests will never get a newspaper!  Hang on…”  He now starts zig-zagging across the corridor, putting a newspaper on one side, then the other, slowly progressing so that each door on both sides has one.  ”Now I’ll make it to every door”, he says.

Feeling a little overwhelmed, you go to the bar and order a gin and tonic.

“How much gin would you like in this glass before I top it up with tonic?” asks the bartender.

“Well, the glass is 10cm tall, so I’ll go with 3cm of gin I reckon” you sigh.

“If I might be so bold, sir, I recommend π cm of gin.  In my experience that gives the optimum taste.”

“Sure” you say and he pours you the drink.  He adds a very, very thin sliver of lime.

“You know, last week, everyone in the hotel came through this bar and they each ordered a different strength of gin and tonic.”

You’re not surprised.  ”Let me guess”, you say “they each ordered a different strength but still there are strengths you’ve never served?”

“Oh yes” he says. “Not only that, but most mixtures were never picked.  In fact, if you were to pick a mixture at random, you’d certainly pick one which had never been served to a guest before.  In fact…” you take a long drink “…even if every guest came in here every day for as long as you like, that would still be true.”

You ask for another drink.

As well as demonstrating some of the weird properties of countable infinities, this story hints at another, even weirder concept. In mathematics, even when looking at just one concept — the sizes of sets — there are different infinities, and some are bigger than others.  These are cardinal numbers.  The first infinite cardinal is called “aleph-null”, and it’s the size of the set of natural numbers.  A bigger cardinal is called “aleph-one”, and is equal to 2 to the power of aleph-null and this is the size of the set of real numbers.  I is also the size of the set of real numbers between any to given real numbers.  This is why there are, in a very real sense, many more mixes of gin and tonic (pretending that gin’s not made of molecules) than there are people in the hotel.  Related to cardinal numbers are ordinals.  All fascinating topics in set theory and logic.  If you find these interesting, the Wikipedia articles are a bit dense but there’s plenty of other resources on the web.  Otherwise, any undergraduate text on set theory should cover these in plenty of detail.

Suicide us the topic for Episode #3 of the Sci Phi Show. Thanks for tuning in. I am making use of the Star Trek Voyager Episode 2×18 called Deathwish about a suicidal Q.

Podcast: Play in new window | Download (Duration: 32:02 — 16.0MB)

Show Notes

• Noodle MX the network I am affiliating with.


• The Standford Encyclopedia of Philosophy contains a great section on suicide. It is long but a good overview of the topic.

• Wesley J Smith has a blog called Second Hand Smoke that looks at all sorts of bioethics issues including euthanasia.
• The clip was taken from Star Trek Voyager Season 2, specifically the episode Death Wish

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Thanks for tuning into episode #2 of the Sci Phi Show. In this episode I talk about “Time & Time Travel”, using Futurama & Star Trek voyager as some source material. I hope you enjoy it.

Podcast: Play in new window | Download (Duration: 24:39 — 12.9MB)

Show Notes

• Noodle MX the network I am affiliating with.


• As I said, the Standford Encyclopedia of Philosophy contains a great section on time as well as a good section on Time Machines
• For follow up reading, there is a section called Reductionism and Platonism with Respect to Time that looks at the two different possible relationship of time to events.
• There is also a good section on J.M.E McTaggart’s arguments about the nature of time which leads into a discussion of the The A Theory and The B Theory of time.
• After that there is a section on the The 3D/4D Controversy and a section on the The Topology of Time to explore.
• Once you’ve done that you can dive right into the discussion of Time Travel
• There is also a good discussion of Time Machines as well as Time Travel and Modern Physics that goes into more depth than I was able to in the show.
• The Grandfather Paradox audio was taken from Futurama, specifically the episode Roswell that Ends Well
• The other clip was taken from Star Trek Voyager, specifically from episode 3×08 Futures End, Part 1.
• Stephen Baxter wrote a collection of short stories called Vacuum Diagrams that contain a number of stories dealing with time travel. It is well worth the read

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Thanks for tuning in the first episode of the new look Sci Phi Show. In this episode I talk about “The Jedi, The Sith and the lies they tell”. I hope you enjoy it.

Podcast: Play in new window | Download (Duration: 17:57 — 9.1MB)

Show Notes

• Rifftrax
• The definition of lying and deception at the Stanford Encyclopedia of Philosophy. A good if longish read.
• The philosopher Immanuel Kant at the Stanford Encyclopedia of Philosophy. Some background if you don’t know who he is.
• Kant’s Moral Philosophy including the Categorical Imperative
• Rahab over at Wikipedia. The article is pretty good

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