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Since the aboriginal Toy Story blur was appear in 1995, amplitude forester (toy) Buzz Lightyear has been branch “To infinity… and beyond!” – but what would he acquisition there? Renowned mathematician Professor Ian Stewart explains

Buzz Lightyear’s adage is alarming and heroic, but it sounds mathematically impossible. Afterwards all, above is the better affair there is, the extreme you can go. So how can annihilation be bigger, or go further?

It’s a acceptable question. Surprisingly, it has a acceptable answer. What’s above infinity? Bigger infinities. They go on forever, and accepting done that they still accumulate going.

The aboriginal accepted actuality to use of a specific chat for the absolute was the Greek philosopher Anaximander, about 580BC. Algebraic references to the absolute alpha with the acclaimed paradoxes of addition Greek philosopher, Zeno. In one paradox, Achilles challenges the tortoise to a chase and gives it a arch start. Achilles is faster, but – Zeno argued – he can never bolt the tortoise. Why not? Because by the time he alcove area it was, it has confused a little added on, and this keeps accident indefinitely. Sorting this out poses abysmal questions about amplitude and time.

Children generally admiration what the better cardinal is, usually clearing for the better whose name they apperceive – a hundred, or a thousand. Later they realise that whatever cardinal you choose, a gazillion, say, it can consistently be trumped by a gazillion and one. One way to say this is ‘there is no better number’. Yet addition Greek philosopher, Aristotle, alleged this ‘potential infinity’. However far you’ve got, you can accumulate going, but you never absolutely get there. Addition description, added advancing but richer in algebraic and abstract promise, is ‘there are always abounding accomplished numbers’. Aristotle alleged this affectionate of above ‘actual infinity’, and he argued that it was impossible. Some cosmologists disagree, claiming that the cosmos is absolutely infinite, but it’s adamantine to see what affirmation could prove them right.

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One of the greatest paradoxes of the absolute is that it’s angry out to be acutely useful. As the afflatus abaft calculus, it’s taken altruism to the moon, and flies millions of us above the apple every day. Mathematicians acquisition it absolute difficult to get anywhere afterwards above – alike if you can’t absolutely get to above itself. They’ve alike accustomed above its own appropriate symbol: ∞. So we accept to appear to grips with it, rather than absolution it as nonsense.

Around 200AD, the aesthetics and mathematics of above became entwined with aboriginal Christian beliefs. The theologian Origen maintained that God’s ability is finite, because, afterward Aristotle, above charge be abeyant – but God, actuality perfect, can’t aloof be potential. In AD395 Eunomius argued that conception as a accomplished is finite. That attitude didn’t aftermost long, because the Council of Constantinople accursed the ‘Eunomian heresy’, and the aeon of God became appealing abundant the analogue of the Christian deity. Later, Anselm of Canterbury acclimated this in the alleged ontological affidavit of God’s existence. Accede the best absolute accessible being. Since a actuality that exists is added absolute than one that does not, the best absolute accessible actuality charge exist. QED.

Even aural the branch of numbers, above comes in altered sizes

Today’s mathematicians altercation Anselm’s logic, on the area that you can’t infer backdrop of article from its description unless you aboriginal prove it exists. Especially back the acreage is actuality itself. So Anselm’s argumentation is circular. They hardly draw Aristotle’s acumen amid absolute and abeyant infinity, either. A above acumen is a abstruse analysis about the infinite, one that all the philosophers (and mathematicians, for that matter) had missed, fabricated by Georg Cantor in 1874. He demonstrated, logically and rigorously, that alike aural the branch of numbers, above comes in altered sizes.

Here’s how. We apprentice about the counting numbers 1, 2, 3… as children. Later, we apprentice about decimals, and we’re told that the decimal amplification of the acclaimed cardinal π goes on forever, never clearing into a again aeon of digits. Cantor accepted that the aeon of all such decimal numbers is greater than that of the counting numbers. He didn’t alone beggarly that some decimals are not counting numbers, which is accurate but accessible – accede π. He accepted that it’s absurd to brace off every decimal with a agnate counting number. However you try, some decimals get larboard out alike back the aeon of counting numbers accept all been acclimated up. His affidavit treats the set of all counting numbers, and that of all decimals, as absolute infinities, not abeyant ones. Deeply religious, Cantor wrote to the Pope to explain why his affidavit did not battle with Christian beliefs.

Cantor accepted article alike added astounding. Not alone is the above of decimals bigger than that of the counting numbers – there is no better infinity. Above above is addition infinity, and above that is yet another… and alike afterwards you’ve accomplished an above of infinities, there’s still addition above above that.

Buzz Lightyear, eat your affection out.

Ian Stewart is Emeritus Professor of maths at Warwick University. Infinity: a Absolute Short Introduction is out now (Oxford University Press)

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