Infinitely More

Infinitely More explores the mathematics and philosophy associated with the concept of infinity, covering a diverse array of topics such as surreal numbers, lattices, the continuum hypothesis, transfinite numbers, mathematical structures, and paradoxes. It examines both the abstract theories and their practical implications, integrating logical analysis and historical perspectives.

Mathematics Philosophy Logic Number Theory Algebra Graph Theory Mathematical Induction Epistemology

Top posts of the year

And their main takeaways
41 implied HN points β€’ 06 Jan 24
  1. The surreal numbers unify various number systems into one comprehensive system.
  2. Surreal numbers are generated through a recursive process of completion and ordering.
  3. The surreal number generation rule involves separating existing numbers into lower and upper sets to create new numbers.
33 implied HN points β€’ 17 Jan 24
  1. A lattice is an order relation where every pair of elements has a least upper bound and a greatest lower bound.
  2. In lattices, the join of two elements is the larger of them and the meet is the smaller of them.
  3. Every linear order, set of positive integers, Boolean algebra, and field of sets can be considered lattices.
30 implied HN points β€’ 11 Jan 24
  1. The eventual domination order involves comparing functions in a specific way
  2. In the eventual domination order, every countable sequence of functions is strictly bounded
  3. This order prohibits one from climbing a simple ladder to reach the top
17 implied HN points β€’ 04 Feb 24
  1. There is a rich algebra of orders involving operations like addition and multiplication.
  2. The disjoint sum operation creates a combined order without interactions between the two parts.
  3. The ordered sum operation combines two orders by placing one above the other, creating new orders with distinct properties.
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15 implied HN points β€’ 02 Mar 24
  1. A formal language for first-order predicate logic involves understanding the basic syntax, terms, variables, and structure interpretations.
  2. Signatures in structures specify the elements like relations, functions, and constants in a mathematical structure, detailing their features and meanings.
  3. Mathematics uses a wide array of first-order structures to study various concepts like orders, graphs, groups, and more, unifying different mathematical investigations.
15 implied HN points β€’ 24 Feb 24
  1. With first-order logic, subtle features can help distinguish mathematical structures from similar alternatives.
  2. Different mathematical structures can be differentiated by how symbols are interpreted in each structure, revealing unique properties.
  3. Finding statements in the language of orders that are true in one structure and false in others can help distinguish mathematical structures.
15 implied HN points β€’ 31 Mar 23
  1. Giants in folklore, acting in a humanlike manner but at a larger scale, are physically impossible according to Galileo.
  2. Galileo's paradox of the giant involves the concept of scaling and how larger objects may not behave as expected when scaled up.
  3. Observations on scaling in different dimensions can lead to various paradoxes of dimension.
12 implied HN points β€’ 19 Feb 24
  1. First-order predicate logic provides a formal language and semantics capable of expressing fine distinctions and shades of meaning.
  2. Understanding quantifiers, such as βˆƒ and βˆ€, is crucial in first-order logic as they allow one to make statements like 'there is an x such that Ο†' or 'every x has property Ο†.'
  3. Engaging in logic puzzles and practice can help in developing a deeper comprehension of first-order logic concepts and their applications.
10 implied HN points β€’ 10 Feb 24
  1. A countable random graph is a graph where you flip a coin to decide the edges between vertices in an infinite set, and the result is the same graph almost every time.
  2. Graph theory is a complex subject with beautiful theorems, and different notions of graphs exist, such as directed graphs and simple graphs.
  3. In mathematics, there are variations in graph definitions, such as allowing reflexivity or multiple edges, but in simpler contexts, graphs are typically referred to as simple graphs.
10 implied HN points β€’ 30 Jan 24
  1. Mathematical induction is a fundamental principle in mathematics, used to prove many fundamental facts in arithmetic and number theory.
  2. The common induction principle states that if a set of natural numbers contains 0 and whenever n is in the set, n+1 is also in the set, then every natural number is in the set.
  3. Strong induction allows the induction step to use multiple smaller numbers to prove a statement, and can be proven from the least-number principle.
5 implied HN points β€’ 07 Mar 24
  1. Truth in a structure is defined by recursion on sentences, reducing to assertions with temporary assignments of variables
  2. The concept of valuation in a model involves assigning variables to specific individuals, treating them as constants in that context
  3. Tarski's disquotational theory of truth posits that an assertion is true when the proposition it asserts is true, forming the foundation of a compositional theory of truth
5 implied HN points β€’ 13 Mar 24
  1. Induction is about the impossibility of minimal counterexamples, and it comes in various forms like common induction and strong induction.
  2. Flexible use of induction is key - choose the valid form that best fits your proof.
  3. Differentiate between examples and proofs - examples can provide insight but don't prove universal statements.
3 HN points β€’ 14 Apr 23
  1. Mathematicians and philosophers may disagree on the nature of existence of infinite collections or infinite objects.
  2. According to potentialism, natural numbers are potentially infinite, allowing for more to be added continuously.
  3. Consider exploring potentialism and actualism for different perspectives on the concept of infinity.