The question of whether there exists a size larger than infinity has fascinated mathematicians and philosophers for centuries. Infinity, represented by the symbol ∞, is an abstract concept referring to something without bound or end. At first glance, it would seem nonsensical to speak of anything being “larger than infinity,” since infinity represents an unlimited amount. However, different types of infinities exist in mathematics, some of which are provably larger than others. This leads to the intriguing question: Is there an “absolute infinity” that is larger than all other possible infinities?
In the late 19th century, German mathematician Georg Cantor revolutionized our understanding of infinity and infinite sets. He showed that some infinite sets are “larger” than others, disproving the ancient notion that all infinities are the same size. Cantor introduced cardinal numbers as a way to compare the sizes of infinite sets. He showed, for instance, that the set of natural numbers {1, 2, 3, …} is smaller than the set of real numbers, even though both sets are infinite.
This insight leads to the question of whether there exists an absolute infinity – an infinity larger than any other possible cardinal number. In 1938, mathematician Paul Cohen proved that the existence of such an absolute infinity is actually independent of the standard axioms of set theory. This means that the question can neither be proved nor disproved logically.
Some mathematicians have hypothesized that an absolute infinity does exist, referring to it as the “Aleph numbers.” The Aleph numbers extend the cardinal numbers into the transfinite. Aleph-null represents the smallest infinity, the size of the natural numbers. Aleph-one is the next largest, then Aleph-two, and so on, extending into the unknown. But is there an Aleph number greater than all others? This remains an open question.
The Different “Sizes” of Infinity
To better understand the notion of some infinities being “larger” than others, let’s look at the different types of cardinal numbers used to measure infinite sets:
Aleph-null (א0) – The cardinality (number of elements) of the set of natural numbers {1, 2, 3…}. This represents the “smallest” kind of infinity.
Aleph-one (א1) – The next largest cardinal after א0, it represents the cardinality of the set of real numbers. This is provably larger than א0.
Continuum hypothesis – States that there is no cardinal number between א0 and א1. This has been shown to be impossible to prove or disprove from the standard ZF axioms of set theory.
Inaccessable cardinals – Cardinals beyond א1 that are larger than anything that can be constructed from below. Their existence cannot be proved or disproved within ZF.
Large cardinals – Refers to cardinals with various “largeness” properties. Their existence would imply there are infinite hierarchies of infinities of different sizes.
Inconsistent multiplicity – The notion that every potential cardinal number corresponds to an actual infinite set. This is impossible to prove or disprove, but implies an unlimited sequence of larger and larger infinities.
So while א0 is the smallest infinity, there is provably a larger cardinal (א1), and modern set theory has shown there is no limit to how large cardinal numbers may become before reaching an absolute infinity, if one exists. This leaves open the possibility of infinite hierarchies of ever-larger infinities.
Arguments For and Against an Absolute Infinity
The possibility of an absolute infinity greater than all others remains a topic of debate among mathematicians. Here are some key arguments on both sides of the issue:
Arguments for an Absolute Infinity
– The set of all ordinal numbers may itself constitute an absolute infinity beyond all cardinal numbers. Ordinals measure the order-type of well-ordered sets, rather than just their cardinality.
– An absolute infinity could exist without contradicting ZF axioms or being provably disprovable from them. It may simply be independent of ZF theory.
– Conceptually, it seems plausible that there should exist some “maximal” infinity if infinite hierarchies are possible. This is analogous to how transfinites like א1 and א2 extend the sequence of finite ordinals.
Arguments against an Absolute Infinity
– No specific mathematical example of an absolute infinity has been constructed. Its existence remains hypothetical.
– Disproving the existence of larger cardinals proves consistency of ZF axioms. This suggests absolute infinities are unlikely to exist based on current foundations.
– Accepting an actual absolute infinity requires a strong metaphysical commitment beyond just mathematics. It may be a philosophical rather than mathematical notion.
– Set theory constructs like the von Neumann universe show no “limit” to the cumulative hierarchy of infinite sets. This suggests no maximal infinity.
Set Theory and the Continuum Hypothesis
The search for an absolute mathematical infinity is closely tied to set theory and Cantor’s famous Continuum Hypothesis (CH). To understand how these are related, let’s look at some key ideas:
– CH states there is no cardinal number between א0 and א1, where א1 represents the cardinality of the continuum (the set of real numbers).
– In 1963 Paul Cohen proved CH is actually independent of ZF axioms. Therefore, it can be neither proved nor disproved from current set theory.
– Gödel had previously shown it is consistent for CH to be true. Cohen showed you could also assume CH false without creating inconsistencies.
– CH remains unresolved despite being such a central question about the nature of infinity. An absolute infinity could provide a definite answer.
– CH also highlights that definitions of infinity depend deeply on what set theory axioms are assumed. This relates to the Platonic versus formalist views of mathematics.
– Some large cardinal axioms imply CH is false. But large cardinals themselves have unclear ontological status within set theory.
Resolving CH and definitively answering questions about larger infinities may require stepping outside ZF foundations to adopt new axioms. The search for an absolute infinity reflects mathematicians’ continued conceptual unease with infinite sets.
Does Infinity Exist in Physical Reality?
Much of the mathematical work on comparing infinities relies on axiomatic set theory. But an important question is whether actual infinities can exist in physical reality. This depends on one’s philosophical perspective:
Finitist view
– Only accepts finite mathematical objects and processes as physically real. Actual completed infinities are considered abstract notions with no real existence.
Potential infinite
– Infinite processes like counting can be realized sequentially, but “actual” infinities cannot be said to be completed. Hence potential, not actual, infinities exist.
Platonism
– Abstract mathematical entities have their own reality apart from physical existence. All cardinalities exist in this Platonic realm of mathematics.
Mathematical universe
– Under the mathematical universe hypothesis, physical reality is fundamentally mathematical. If so, its intrinsic infinity should match that described by mathematics.
Absolute infinities in physics?
– Modern physics offers some hints of infinities in nature, like spacetime singularities and quantum field theory vacuums. But their detailed characteristics remain elusive.
It is not clear whether infinity in the concrete physical world matches the diverse cardinalities that axiomatic set theory produces. The existence and accessibility of absolute infinities in physics remains speculative. Philosophical inclination toward Platonic realism or nominalism flavors one’s perspective here.
Logical and Philosophical Issues with Absolute Infinities
The notion of a largest possible cardinal number raises some logical and philosophical concerns:
Self-referential paradoxes
– An absolute infinity must contain itself as a member, leading to Russell’s paradox. This suggests unrestricted comprehension of an absolute set leads to contradiction.
Definability Issues
– An absolute infinity would have to be defined as something like “the set of all ordinals,” But explicit definition may be inherently impossible.
Conceptual vagueness
– What does it mean to be larger than all other conceivable infinities? This relies on a concept of “conceivability” that is hard to formalize mathematically.
Unprovable existence
– Consistency of ZF axioms implies absolute infinities are unlikely to exist within standard set theory. Their hypothesized existence may simply reflect wishful thinking.
These issues suggest the notion of an absolute mathematical infinity may encounter logical barriers or exist only as a philosophical concept, not a formal mathematical entity. But exploring these barriers provides insight into the scope and limits of infinity and set theory itself.
Absolute Infinities in Philosophy and Mysticism
Speculation about absolute infinities extends beyond just mathematics into metaphysics and spirituality:
The Absolute (Brahman)
– In Hindu philosophy, Brahman refers to an infinite absolute reality beyond all concepts and limitations. This bears similarities to an absolute infinity.
The Godhead
– Mystical Christian theology conceives of an infinite and unconditioned Godhead beyond the Trinity conception of God. This evokes parallels to a highest form of infinity.
The One (Neoplatonism)
– Neoplatonic philosophy described a supreme, infinite One beyond being and knowledge. This highest unity resembles an absolute infinity.
The Infinite (F.W.J Schelling)
– Philosopher Schelling conceived of an endless, unbounded Absolute which cannot be reduced to a finite concept. It aligns with an absolute mathematical infinity.
In these metaphysical contexts, the absolute infinity represents a transcendent, unqualifiable totality – a limit to conceptualization itself. Mathematicians’ intuitions of a largest conceivable infinity may stem from similar primal human notions of the unlimited Absolute. The desire to reach infinity’s “end” reflects mystical yearnings to attain the infinite divine source.
Conclusion
The question of whether an absolute mathematical infinity exists remains open-ended. Modern set theory has shown there are “larger” infinities beyond any given cardinal, but has not revealed a definite endpoint. The existence of a largest conceivable infinity currently can be neither proved nor disproved based on accepted axioms. Intuitions of an absolute infinity seem linked to mystical and philosophical notions of a Totality or Absolute. Pursuing the absolute infinite drives exploration of set theory’s outer limits, but may ultimately reflect metaphysical beliefs more than mathematical certainties. The question “Is there a biggest infinity?” thus crosses over into profound issues at the foundations of mathematics, philosophy, and humanity’s conception of the infinite cosmos.