The Social Security Trust Fund does exhaust it's surplus (if no additional changes are made to the system), but that reduces benefits to ~3/4 of those promised; benefit payments do not disappear entirely.

"In the annual Trustees Report, projections are made under three alternative sets of economic and demographic assumptions. Under one of these sets (labeled "Low Cost") the trust funds remain solvent for the next 75 years. Under the other two sets (the "Intermediate" and "High Cost"), the trust funds become depleted within the next 20 years. The intermediate assumptions reflect the Trustees' best estimate of future experience.

Some benefits could be paid even if the trust funds are depleted. For example, under the intermediate assumptions, annual income to the trust funds is projected to equal about seventy-nine percent of program cost once the trust funds become depleted. If no legislation has been enacted to restore long-term solvency by that time, about three-quarters of scheduled benefits could be paid in each year."

As long as the government continues to exist and has the ability to tax, Social Security will be around.

https://www.ssa.gov/oact/progdata/fundFAQ.html

https://www.ssa.gov/oact/solvency/index.html