BMPJ: The Infinities Saga (Part 0: The Last Increment to The Finality)

Instructions

Explanation for each number: The Last Increment (TLI): The final number that can be subtracted from to get a different result. Its value is realistically infinite, but it behaves as if it is finite. Natural (ℕ): The sum of every positive whole number in a base-ten system. Because of this, it's basically the (The Last Increment)th triangle number/step squad. Finity (∝): The sum of every single finite number ever written or thought of. This includes numbers such as Natural, The Last Increment, or even Finity itself. Thinking of Finity practically doubles Finity. Thinking of modifications of Finity (such as TREE(Finity) or FOOT^Finity(Finity)) can propel Finity to much higher values. Second-Powered Finity (⨍): According to Retr0r0bl0x1anCR of the FG wiki (https://fictional-googology.fandom.com/wiki/Second-Powered_Finity), Second-Powered Finity is, and I quote, "((∝ Cycle ∝) Phase ∝)^∝" (reminder that ∝ is Finity). I have no idea what cycles or phases are (in this context, at least). This number is, of course, far larger than Finity. Strong Finity (its symbol becomes a ? in project descriptions, so I'll write Strong Finity as F in this description): This number has two definitions according to https://fictional-googology.fandom.com/wiki/Strong_Finity. One definition makes Strong Finity equal to "Tar(TTDH(SSCG(TREE({⨍, ⨍, ⨍} & ⨍))))" (reminder that ⨍ is Second-Powered Finity). The other definition defines Strong Finity as "⨍{{{{{⨍}}}}}⨍ & ⨍{{{{{⨍}}}}}⨍" (The ampersand (&) is the "array of" operator as explained in https://googology.fandom.com/wiki/Array_of) (another reminder that ⨍ is Second-Powered Finity). Of course, both of these notations exceed Second-Powered Finity. Stronger Finity (similar to Strong Finity, Stronger Finity's symbol will become a ? when typed in this description, so I'll refer to Stronger Finity as G): Stronger Finity, similar to Strong Finity, has two different definitions (see https://fictional-googology.fandom.com/wiki/Stronger_Finity). The first definition is "[{F(ℕ) & 10^100, ⨍, ⨍}^{F(ℕ) & 10^100, ⨍, ⨍}]x∝" (reminder that F is treated here as Strong Finity, ℕ is Natural, ⨍ is Second-Powered Finity, and ∝ is Finity). The second definition is "F & F & F & F & … F & F" (Reminder that F is being treated as Strong Finity here and the ampersands (&) are "array of" operators.) with F F's. This can be written as F && F according to Retr0R0bl0x1an's Repeated Array of Operator (https://fictional-googology.fandom.com/wiki/Retr0R0bl0x1an%27s_Repeated_Array_of_Operator). Absolute Finity (ᴒ or ဂ): According to Retr0R0bl0x1an, Absolute Finity is equal to "Ipsius Number (this number is explained later) - (Finity/2)/2/2" (https://fictional-googology.fandom.com/wiki/Absolute_finity,). Ipsius Number: (This number is typically placed after Ifinity, but Absolute Finity's definition makes Ifinity objectively far bigger than this number). Ipsius Number marks the end of countable numbers and the beginning of uncountable numbers. Ifinity (⧜): Not to be confused with Infinity, Ifinity is equal to "{Absolute Finity (Retr0R0bl0x1an's Definition)^Finity + Natural, Strong Finity, Stronger Finity, Second-Powered Finity}" (https://fictional-googology.fandom.com/wiki/Ifinity). The Finality (∠): The Finality is described as the final finite number (https://fictional-googology.fandom.com/wiki/%E2%88%A0_The_Finality). Practicality-wise, it may as well be infinite. But it's not quite at the point of infinity.